Optimal Linear State Feedback Control Research Articles

Indefinite LQ optimal control with cross term for discrete‐time uncertain systems

Recently, there has been an increasing interest in the study on uncertain optimal control problems. In this paper, a linear quadratic (LQ) optimal control with cross term for discrete‐time uncertain systems is considered, whereas the weighting matrices in the cost function are allowed to be indefinite. Firstly, a recurrence equation for the problem is presented based on Bellman's principle of optimality in dynamic programming. Then, a necessary condition for the existence of an optimal linear state feedback control of the indefinite LQ problem is given by the recurrence equation. Moreover, a sufficient condition of well‐posedness for the indefinite LQ problem is presented by introducing a linear matrix inequality (LMI) condition. Furthermore, it is shown that the well‐posedness of the indefinite LQ problem, the solvability of the indefinite LQ problem, the LMI condition, and the solvability of the constrained difference equation are equivalent to each other. Finally, an example is presented to illustrate the results obtained.

Optimal Periodic-Gain Fractional Delayed State Feedback Control for Linear Fractional Periodic Time-Delayed Systems

This paper develops the fundamentals of optimal-tuning periodic-gain fractional delayed state feedback control for a class of linear fractional-order periodic time-delayed systems. Although there exist techniques for the state feedback control of linear periodic time-delayed systems by discretization of the monodromy operator, there is no systematic method to design state feedback control for linear fractional periodic time-delayed (FPTD) systems. This paper is devoted to defining and approximating the monodromy operator for a steady-state solution of FPTD systems. It is shown that the monodromy operator cannot be achieved in a closed form for FPTD systems, and hence, the short-memory principle along with the fractional Chebyshev collocation method is used to approximate the monodromy operator. The proposed method guarantees a near-optimal solution for FPTD systems with fractional orders close to unity. The proposed technique is illustrated in examples, specifically in finding optimal linear periodic-gain fractional delayed state feedback control laws for the fractional damped Mathieu equation and a double inverted pendulum subjected to a periodic retarded follower force with fractional dampers, in which it is demonstrated that the use of time-periodic control gains in the fractional feedback control generally leads to a faster response.

Optimal, Combined Speed, and Direct Thrust Control of Linear Permanent Magnet Synchronous Motors

An optimal control scheme synthesized using a linear quadratic regulator-based approach for combined speed and direct thrust force control of permanent magnet synchronous motor (PMSM) is proposed. A novel multiple-input-multiple-output state space model, having stator flux, thrust force, and mover's speed as states, is formulated for the linear PMSM where the state transition matrix is independent of mover's speed. An optimal linear state feedback control scheme is designed using the optimal linear quadratic regulator technique. Integral action is added to the designed control scheme by state augmentation to minimize the steady-state error. Experimental results clearly prove that the proposed optimal control scheme results in a faster transient response of speed and force with improved steady-state regulation of force and flux when compared with the state of the art.

Indefinite LQ optimal control with equality constraint for discrete-time uncertain systems

Based on uncertainty theory, this paper studies a kind of discrete-time uncertain linear quadratic (LQ) optimal control with equality constraint for the terminal state, allowing the state and control weighting matrices in the cost function to be indefinite. First, we transform the uncertain LQ optimal control problem into an equivalent deterministic optimal control problem. Then, a necessary condition for the existence of optimal linear state feedback control is presented by means of matrix minimum principle. Moreover, the well-posedness of the uncertain LQ problem is proved by applying the technique of completing squares. Finally, an example is provided to demonstrate the effectiveness of our theoretical results.

A Linear Quadratic Regulator-Based Optimal Direct Thrust Force Control of Linear Permanent-Magnet Synchronous Motor

Linear permanent-magnet machines are often characterized by low inductance and short pole-pitch which leads to a small operational range of load angles. The resultant control performance using conventional direct thrust force control (DTFC) techniques is poor with high force ripple. This research improves this aspect of DTFC. A novel multiple-input multiple-output (MIMO) state-space model, independent of the mover’s speed, having stator flux and thrust force as states, is formulated for the linear permanent-magnet synchronous motor (PMSM). An optimal linear state feedback control scheme is then designed using the optimal linear quadratic regulator technique. Integral action is added to the designed control scheme by state augmentation to minimize the steady-state error and reduce the force ripple. Experimental results clearly prove that the proposed optimal control scheme results in a faster transient response of speed and force with improved steady-state regulation of force and flux when compared to the state of the art.

An online optimal linear state feedback controller based on MLS approximations and a novel straightforward PSO algorithm

Selection of control parameters is an important issue in the field of control design. This selection depends on the initial condition of the systems. On the other hand, controller parameters should be adjusted under all initial conditions to reach the optimal performance. To overcome this problem, in this paper, an online optimal linear state feedback control is introduced. Firstly, a straightforward particle swarm optimization is used to optimize the state feedback control parameters under some certain initial conditions. Then, in order to adapt the optimal controller to different initial conditions, the moving least squares approximation is used. The proposed technique is applied to an inverted pendulum system. The feasibility and efficiency of the proposed controller are assessed in simulation results.

Study on Indefinite Stochastic Linear Quadratic Optimal Control with Inequality Constraint

This paper studies the indefinite stochastic linear quadratic (LQ) optimal control problem with an inequality constraint for the terminal state. Firstly, we prove a generalized Karush-Kuhn-Tucker (KKT) theorem under hybrid constraints. Secondly, a new type of generalized Riccati equations is obtained, based on which a necessary condition (it is also a sufficient condition under stronger assumptions) for the existence of an optimal linear state feedback control is given by means of KKT theorem. Finally, we design a dynamic programming algorithm to solve the constrained indefinite stochastic LQ issue.

The (multi-player) linear quadratic state feedback control problem for index one descriptor systems

This paper gives a complete characterization of all optimal linear quadratic static state feedback controllers if the system is described by an index one descriptor system E x ˙ ( t ) = Ax ( t ) + Bu ( t ) . No definiteness restrictions are made with respect to the quadratic performance criterion. The explicit characterization of all solutions is achieved by transforming to a coordinate system, which can be computed by performing a singular value decomposition of E. This enhances the formulation of a numerical algorithm to calculate the solutions. Based on this explicit characterization of the solution set, we can characterize the set of all linear feedback Nash equilibria, in the context of non-cooperative linear quadratic differential games. The results are illustrated in an example.

Linear and non‐linear simulations of feedback control in plane Poiseuille flow

AbstractThis paper examines the performance of optimal linear quadratic state and output feedback controllers in stabilizing two‐dimensional perturbations in a plane Poiseuille flow. The synthesis of the controllers is based on a linearized model of the flow using a new set of interpolating polynomials in the wall‐normal direction, which automatically satisfy the homogeneous Dirichlet and Neumann boundary conditions at the walls and eliminate spurious eigenvalues. The controllers are implemented into a non‐linear Navier–Stokes solver, which is modified to compute the evolution of the flow perturbations. Two cases are examined, one with small initial disturbances that do not violate the linearity assumptions and the other with much larger disturbances that trigger the non‐linear convection terms. For the smallest disturbances, the solver accurately reproduced the results of the linear simulations of open‐ and closed‐loop systems. The simulations for the larger disturbances without control showed a rapid initial growth but the flow soon reached a saturated state in agreement with previous findings in the literature. The large initial growth is a consequence of the non‐normal nature of the system dynamics. The state feedback and output feedback controllers were able to reduce significantly the perturbation energy. For the larger disturbances, the energy calculated from the state variables is well below the energy evaluated by direct integration of the velocity field. This is probably due to the non‐linear terms transferring energy to harmonics of the considered wavenumber, which are not sensed by the linear controller. Copyright © 2008 John Wiley & Sons, Ltd.

Optimal control of precision paraboloidal shell structronic systems

Paraboloidal shells of revolution are commonly used in advanced aerospace, civil and telecommunication structures, e.g., antennas, reflectors, mirrors, rocket fairings, nozzles, solar collectors, dome structures, etc. A structronic shell system is defined as an elastic shell embedded, bonded or laminated with distributed piezoelectric sensors and actuators and it is governed by either in situ or external control electronics. A closed-loop control system of paraboloidal shell structronic system consists of distributed sensors/actuators and controller coupled with an elastic paraboloidal shell. State equation for the paraboloidal shell structronic system is derived and optimal linear quadratic state feedback control is implemented, such that the “best” shell control performance with the least control cost can be achieved. The gain matrix is estimated based on minimizing a performance criterion function. Optimal control effects are compared with controlled responses with other non-optimal control parameters. Control effects of identical-sized sensor/actuator patches at different locations are studied and compared. Modal control effects for different natural modes are also investigated.

On static feedback for the ℓ 1 and other optimal control problems

Although Ê 1 -optimal linear state feedback controllers are known to be dynamic, it has been shown that suboptimal performance arbitrarily closed to optimal can be achieved by using a static non-linear feedback law. In this paper, this fact is established by using a novel approach which shows that the result is a natural consequence of elementary state-space theory. The approach is motivated by recent works in active vision systems, which have considered a state-feedback problem tightly connected with Ê 1 optimization. This problem, which has independent interest, is discussed in some detail. The new formulation of the problem provides additional insight in Ê 1 state-feedback. In particular, it leads naturally to some extensions which do not follow in a straightforward manner from previous works on the subject.

Output feedback controllers of suboptimality degree beta

A misconception in a recent paper by D.Z. Zheng (1989) about suboptimal feedback control is highlighted. A novel synthesis method for designing output feedback controllers of suboptimality degree beta is developed. These controllers yield a cost value equal to or less than a scalar multiple ( beta ) times the cost value obtained by optimal linear quadratic state feedback control. The output feedback control gains are explicit in terms of the standard Riccati equation when certain conditions are satisfied. >

Nonlinear control of linear multivariable systems via state-dependent feedback gains

It is often desirable to have controllers which respond fast to large errors and which respond slowly to the small errors that are often due to sensor noise. Some nonlinear controllers having these properties are developed here by modifying optimal linear state feedback controllers so that they have state-dependent gains. A general approach is developed for the nonlinear control of multivariable systems such that the transient responses exhibit the desired properties, and the closed loop systems are asymptotically stable.