Linear State Feedback Control Law Research Articles

Control of an Aeroelastic System with Actuator Saturation

The objective of this study is the semiglobal stabilization of the benchmark active control technology wind-tunnel model developed by NASA Langley Research Center. This study applies established theoretical results to the design of a saturation linear state feedback control law. The framework of this investigation is the construction of the null-controllable region and domains of attraction of the planar antistable subsystem of the higher order model in its canonical modal form. Numerical simulations at different densities of the unstable wind-tunnel system are presented that demonstrate construction and performance of the control law applied to an a priori defined subset of the null-controllable region.

A linear matrix inequality approach for guaranteed cost control of systems with state and input delays

The robust control problem for linear systems with parameter uncertainties and time-varying delays is examined. By using an appropriate uncertainty description, a linear state-feedback control law is found, ensuring the closed-loop system's stability and a performance measure, in terms of the guaranteed cost. A linear matrix inequality objective minimization approach allows to determine the optimal choice of free parameters in the uncertainty description, leading to the minimal guaranteed cost

Optimal persistent disturbance attenuation control for linear hybrid systems

In this paper, the disturbance attenuation properties for a class of linear hybrid systems are investigated, and a hybrid optimal persistent disturbance attenuation control problem is studied. First, a procedure is developed to determine the minimal l ∞ induced gain of linear hybrid systems. However, for general hybrid systems, the termination of the procedure is not guaranteed. Then, the decidability issues are discussed. The termination of the procedure in a finite number of steps is shown for a subclass of hybrid systems with simplified discrete event dynamics, called switched linear systems. Finally, the optimal persistent disturbance attenuation controller synthesis problem is studied. It is shown that the optimal performance level can be achieved by a piecewise linear state feedback control law, and a systematic approach is proposed to design such feedback control.

Extended invariance and its use in model predictive control

The use of time invariant linear state feedback control laws for the definition of terminal invariant regions can be conservative, thereby reducing the efficacy of predictive control in terms of size of stabilisable sets and closed-loop performance. This difficulty, which is particularly pronounced in the case of nonlinear and/or uncertain dynamics, can be remedied through the use of time-varying control laws and terminal invariant sets. In existing MPC schemes, however, these have to be computed online thereby rendering implementation impracticable for anything other than low-dimensional systems. Here, the definition of invariance is extended to apply over ν predicted control moves, thereby enabling the use of pre-determined (offline) time-varying state feedback gains. More importantly, this extension allows for the use of local uncertainty or linear difference inclusion sets, and thus affords significant improvements, e.g. in terms of the size of terminal regions.

Robust input-output energy decoupling for uncertain singular systems

This paper addresses the robust input-output energy decoupling problem for uncertain singular systems in which all parameter matrices except E exist as time-varying uncertainties. By means of linear matrix inequalities (LMIs), sufficient conditions are derived for the existence of linear state feedback and input transformation control laws, such that the resulting closed-loop uncertain singular system is generalized quadratically stable and the energy of every input controls mainly the energy of a corresponding output, and influences the energy of other outputs as weakly as possible.

H∞ DISCRETE TIME FUZZY CONTROL WITH APPLICATION TO CHAOS CONTROL BASED ON PIECEWISE LYAPUNOV FUNCTIONS

This paper presents an H∞ controller synthesis method for discrete time fuzzy dynamic systems based on a piecewise smooth Lyapunov function. The basic idea of the approach is to design a piecewise linear state feedback control law and use a piecewise smooth Lyapunov function to establish the global stability with H∞ performance of the resulting closed loop fuzzy control systems. It is shown that the control laws can be obtained by solving a set of Linear Matrix Inequalities (LMI). Application to control chaotic systems is given to illustrate the performance and advantages of the proposed method.

EXPLICIT HYBRID OPTIMAL CONTROLLER FOR DISTURBANCE ATTENUATION IN LINEAR HYBRID SYSTEMS

In this paper, the disturbance attenuation properties for classes of linear hybrid systems perturbed by exterior disturbances are investigated, and a hybrid l1 robust optimal control problem is studied. First, a procedure is developed to determine the minimal l∞ induced gain of linear hybrid systems. However, for general hybrid systems, the termination of the procedure is not guaranteed. Then, the decidability issues are briefly discussed. Finally, we study the robust l1 optimal controller synthesis problem. It is shown that the optimal performance level can be achieved by a piecewise linear state feedback control law for the linear hybrid systems, and a systematic approach to design such feedback control is proposed.

Robust Regulation of Metro Lines Using Time-Variant Control Law

This paper presents a new formulation for robust traffic regulation of metro lines. The approach uses a linear time-variant state feedback control law computed in real-time. The formulation is based on the stability analysis using the eigenvalues of the feedback system and consider the traffic model uncertainties and the constraints on its state and control variables. It is assumed the constraints on model variables and uncertain parameter domain defined by convex compact polyhedra. The simplicity and computational efficiency of this formulation makes it applicable to real-time regulation of nowadays metro lines and presents better performance than the obtained using already known robust regulation approaches.

Semiglobal Robust Stabilization of Nonlinear Systems without Vanishing Condition

In this paper, the problem of semiglobal asymptotic stabilization is considered for a class of nonlinear systems. Although many control schemes assume a system with a vanishing condition, a linear partial state feedback control law is provided even if uncertainties exist at the origin. All the states of an uncertain system can be semiglobally asymptotically stabilizable by the method of "adding integrator". This design method is suitable to a normal form system.

Memoryless H∞ controller for singular systems with delayed state and control

In this paper, the memoryless H ∞ controllers for singular systems with delayed state and control are presented. Based on the modified Riccati-equation approach, a memoryless linear time-invariant state feedback control law is developed. The controller, which is a delay-independent stabilizer for the singular delayed system will simultaneously reduce the H ∞ norm of the closed-loop transfer function from the disturbance to the controlled output to a prescribed level. An illustrative example is included to demonstrate our proposed approach.

H∞ control for discrete-time linear systems with Frobenius norm-bounded uncertainties

In this paper, we consider the problems of robust stability and control for the class of uncertain discrete-time linear systems with Frobenius norm-bounded parameter uncertainties in all matrices of the system and output equations. Necessary and sufficient conditions for the above problems are proposed. A linear static state feedback control law is designed, which is in terms of a Riccati inequality. The results obtained here show that the robust control problem of the uncertain system is equivalent to the control problem for a corresponding uncertainty-free system.

Robust H/sub /spl infin// control for linear discrete-time systems with norm-bounded nonlinear uncertainties

This paper studies the problem of robust control of a class of uncertain discrete-time systems. The class of uncertain systems is described by a state-space model with linear nominal parts and norm-bounded nonlinear uncertainties in the state and output equations. The authors address the problem of robust H/sub /spl infin// control in which both robust stability and a prescribed H/sub /spl infin// performance are required to be achieved, irrespective of the uncertainties. It has been shown that instead of the nonlinear uncertain system, one may only consider a related linear uncertain system and thus a linear static state feedback control law is designed, which is in terms of a Riccati inequality.

Robust H∞ controller design for time-varying uncertain linear systems with time-varying state and control delays

The robust H∞ control problem for time-varying uncertain linear systems with time-varying state, and control delays is considered. Based on the Riccati equation approach, we design a robust H∞ linear time-invariant state feedback control law, which stabilizes the system and reduces the effect of the disturbance input on the controlled output to a prescribed level for all admissible uncertainties. A sufficient condition for the existence of the. robust H∞ controller is derived and two synthesis methods are presented for a robust H∞ control law to uncertain time-delay systems. One is to construct a robust H∞ control law by an equivalent linear time-invariant structural description for the sufficient condition and the control law can be obtained by solving an algebraic Riccati equation (ARE). The other method is to transform the sufficient condition into a linear matrix inequality (LMF) problem and the control law can be obtained by solving the LMI. A numerical example is presented to illustrate the obtained results.

Constrained control for uncertain discrete-time linear systems

In this paper the problem of stabilizing uncertain linear discrete-time systems under state and control linear constraints is studied. Many formulations of this problem have been given in the literature. Here we consider the case of finding a linear state feedback control law making a given polytope in the state space positively invariant while the control remains bounded within prefixed values under the effect of all the uncertainty sequences belonging to a given polytope in the perturbations space. A necessary and sufficient condition for the existence of a solution of this problem is first given. This condition leads to a set of linear constraints which can be solved using linear programming tecniques by defining an appropriate objective function. A worked example shows the effectiveness of the proposed algorithm. © 1998 John Wiley & Sons, Ltd.

Design of performance robustness for uncertain linear systems with state and control delays

The linear systems considered in this paper are subject to uncertain perturbations of norm-bounded time-varying parameters and multiple time delays in system state and control. The time delays are uncertain, independent of each other, and allowed to be time-varying. The integral quadratic cost criterion is employed to measure system performance. Using solutions of Lyapunov and Riccati equations, a linear state feedback control law is proposed to stabilize the perturbed system and to guarantee an upper bound of system performance, which is applicable to arbitrary time delays.

Robust memoryless H∞ controller design for linear time-delay systems with norm-bounded time-varying uncertainty

We consider the robust H∞ control problem for linear time-delay systems with norm-bounded time-varying uncertainty. Based on the Riccati-equation approach, we present a method for designing a robust memoryless linear time-invariant state feedback control law, which stabilizes the system and reduces the effect of the disturbance input on the controlled output to a prescribed level for all admissible uncertainties.

Memoryless H∞ controller design for linear systems with delayed state and control

We present a memoryless H∞ control design method for linear time-invariant systems that have no parametric uncertainty but delay in the state and control. Based on the Riccati-equation approach, we design a memoryless linear time-invariant state feedback control law, which guarantees the asymptotic stability of the closed-loop control system and reduces the effect of the disturbance input on the controlled output to a prescribed level.

Constrained regulation of linear systems

In this paper the Linear Constrained Regulation Problem without an assigned set of initial states is investigated. This problem consists of the determination of a linear state feedback control law and of a set of admissible initial states D, so that every trajectory of the resulting closed-loop system which emanates from the set converges asymptotically to the origin without violation of the control constraints. By applying well-known results on the positive invariance of polyhedral sets, an eigenstructure assignment approach to this problem is established. If the number of unstable open-loop eigenvalues does not exceed the number of control variables, then this approach enables one to derive a linear state-feedback stabilizing control law which makes the set of states where the control constraints are respected positively invariant. This positively invariant set is the maximal admissible set of initial states for the control law under consideration. For the case where the number of control variables is greater than the number of the unstable open-loop eigenvalues, a modified algorithm is proposed.

Feedback control of time‐delay systems with bounded control and state

This paper is concerned with the problem of stabilizing linear time‐delay systems under state and control linear constraints. For this, necessary and sufficient conditions for a given non‐symmetrical polyhedral set to be positively invariant are obtained. Then existence conditions of linear state feedback control law respecting the constraints are established, and a procedure is given in order to calculate such a controller. The paper concerns memoryless controlled systems but the results can be applied to cases of delayed controlled systems. An example is given.

Continuous-time saturated state feedback regulators: theory and design

A linear continuous-time system with saturation on controls becomes a non-linear closed-loop system when linear state feedback control law is implemented. Assuming that the system to be controlled is stabilizable, the possibilities for an invariant feedback matrix to make the saturated closed-loop system globally asymptotically stable (GAS) are studied. It is proven that this stability property can be obtained by means of a linear feedback provided the considered system is stabilizable with respect to a Lyapunov function of the open-loop system. It is also shown that the global stability property in a closed-loop can only concern asymptotically or critically stable open-loop systems. However, except for a strictly unstable linear open-loop system, i.e. with at least one eigenvalue with a positive real part, there always exists some linear feedback linear matrix for which the stability domain can be as large as desired, even if the open-loop system is critically unstable. In all previous cases, the suitabl...