A gradient flow approach to computing lq optimal output feedback gains

This short communication considers the linear quadratic problem with static output feedback. It is shown that an optimal solution can be successfully computed by finding the limiting solution of an ordinary differential equation which is given in terms of the gradient flow associated with the cost function. Several properties are obtained concerning the gradient flow. For example, it is shown that the flow contains a subsequence convergent to a locally optimal output feedback gain. In the special case of state feedback the flow is guaranteed to converge to the optimal gain. The effectiveness of the method is demonstrated by an example.     

Optimal control of a flexible manipulator and controller order reduction

Presented is a control system design study for a flexible manipulator. The device consists of a DC electric motor drive connected via two flexible links to an end effector for position control. The paper details in tutotrial fashion optimal control designs using both H₂ and H_∞ methods with dynamic weighting. The resulting controller is found to be fairly of high order for implementation and so controller order reduction is considered. It is observed that reduction of the controller order beyond a nominal amount cannot be done. A reason for this is postulated using novel perturbation models where it is found that the controller reduction problem is, in this case, a roadblock to practical implementation of the control. Copyright © 1998 John Wiley & Sons, Ltd.     

A multi-input/multi-output optimal PI controller for redundant robots in the presence of flexible disturbances

The two-stage synthesis of a multi-input/multi-output optimal proportional-integral (PI) controller is described for linear, time-invariant systems. In the first stage the PI controller is designed by solving a steady state algebraic Riccati equation. As a result, the optimal cost is expressed in terms of the system's constant output set-points. In the second stage the cost is further reduced by optimally selecting the output set-points to minimize a static quadratic performance index subject to linear algebraic constraints. The design framework is applied to a planar redundant robotic manipulator equipped with four joints and mounted at the tip of a long flexible arm. We then address the problem of self-motion control in the presence of vibratory disturbances.     

Optimal non-linear feedback regulation of spacecraft angular momentum

Optimal non-linear feedback laws are presented for the regulation of spacecraft angular momentum by using flywheels and by using fixed on-off reaction jets. The control law for the flywheels takes the form of a linear combination of functions of the state, each positively homogeneous of degree 2p-1 (p1/2); whereas the control law for the reaction jets is of bang-bang form. The range (of the initial disturbance) over which the flywheels are capable of regulating the system and the problem of momentum dumping are also discussed with regard to flywheel saturation constraints. Finally, a control scheme combining the merits of reaction jets and flywheels is proposed.     

Comparison of two optimal feedback controls for parabolic systems

Two numerical methods for solving minimum energy control problem with final state constraints are presented for parabolic systems. In infinite dimensions, the functional analysis approach gives an explicit solution from which an approximation method is developed to allow the removal of the indetermination which causes numerical difficulties. The Riccati equation approach about the penalized criterion is also studied and the two solutions are compared on the implemented control laws (in finite dimensions). An example is presented.     

Reference output tracking control for a flexible air‐breathing hypersonic vehicle via output feedback

This paper addresses the problem of reference output tracking control for the longitudinal model of a flexible air‐breathing hypersonic vehicle (FAHV) by utilizing the output feedback control approach. The dynamic characteristics of the FAHV along with the aerodynamic effects of hypersonic flight make the flight control of such systems highly challenging. Moreover, there exist some intricate couplings between the engine and flight dynamics as well as complex interaction between rigid and flexible modes in the longitudinal model. These couplings bring difficulty to the flight control design for the intractable hypersonic vehicle systems. This paper deals with the problem of reference output tracking control for the longitudinal model of the FAHV. By utilizing the trim condition information including the state of altitude, velocity, angle of attack, pitch angle, pitch rate and so on, the linearized model is established for the control design objective. Then, the reference output velocity and altitude tracking control design problem is proposed for the linearized model. The flexible models of the FAHV system are hardly measured because of the complex dynamics and the strong couplings of the FAHV. Thus, by using only limited flexible model information, the reference output tracking performance analysis criteria are obtained via Lyapunov stability theory. Then, based on linear matrix inequality optimization algorithm, the static output feedback controller is designed to stabilize the closed‐loop systems, guarantee a certain bound for the closed‐loop value of the cost function, and can make the control output achieve the reference velocity and altitude tracking performance. Subsequently, the condition of dynamic output feedback controller synthesis is given in terms of linear matrix inequalities and a numerical algorithm is developed to search for a desired dynamic output feedback controller which minimizes the cost bound and obtains the excellent reference altitude and velocity tracking performance simultaneously. The effectiveness of the proposed reference output tracking control method is demonstrated in simulation part. Furthermore, the superior reference velocity and altitude performance commands could be achieved via using static and dynamic output feedback controllers under lacking some unmeasured flexible states information in the measurement output vector. Copyright © 2011 John Wiley & Sons, Ltd.     

Quadratic boundedness of LPV systems via saturated dynamic output feedback controller

For the linear parameter varying systems with bounded disturbance, a saturated dynamic output feedback controller is designed by specifically considering input saturation, to stabilize the closed‐loop system. The controller parameters and the corresponding region of attraction are calculated by solving an off‐line optimization problem with respect to input saturation, state constraints, and robust stability. In the off‐line optimization problem, both the unknown and available scheduling parameters are considered for the linear parameter varying systems. When the unknown scheduling parameters are considered, the off‐line optimization problem is nonconvex and can be solved by the cone complementary linearization method. When the available scheduling parameters are considered, the off‐line optimization problem can be reformulated as convex optimization due to the parameter dependent form of controller parameters. In the both cases, input saturation is specifically handled by introducing a set of linear matrix inequalities into the off‐line optimization problem, which can reduce the conservatism of the controller design and fully exploit the controller capability. Based on the real‐time estimated state, system output, and scheduling parameters, the actual input can be obtained by saturating the dynamic output feedback controller, and steer the augmented state quickly converge to the neighborhood of the origin. Two numerical examples are provided to illustrate the proposed approaches.     

An Optimal output feedback control method and application to a motor-generator system

A multi-input/multi-output feedback control system synthesis method with output feedback structure is proposed on the basis of an improved optimal regulator theory. The method is applied to a DC motor/AC generator system. The effectiveness of the method is demonstrated by simulation and experimental studies. An input-output relation is derived through the observable canonical form of the state equations of the controlled object, and an improved optimal regulator theory is applied to the relation. The controller consists of proportional actions, integral actions, difference actions and compensating actions for input delay times. The (l — 1) th-order difference actions are included in the controller for the controlled object in which an lth-order output subsystem is included. On the other hand, the (n — 1) th-order difference actions are included in the previous method for the same controlled object. In order to consider the transient response of the designed control system, the relations between the weighting factors in the defined performance index and the transient response are studied by simulation and experiment. Satisfactory responses for a two-input/two-output system are obtained in our experiments.     

Synthesis of optimal feedback guidance law for homing missiles using neural networks

Most existing missiles are guided by proportional navigation guidance (PNG) law, but PNG is a particular case for LQ guidance rule with two main assumptions of small line‐of‐sight angles and negligible acceleration along the line‐of‐sight. However, most missile engagements exceed these limits because of high tangential and normal accelerations. Unfortunately, it is not possible to determine the feedback guidance law for non‐linear systems such as homing missiles in real‐time. We use artificial neural networks to synthesize feedback laws for homing missiles with non‐linear state equations. We first obtain an open‐loop optimal numerical solution for non‐linear state equations and then use these data to train a feed‐forward multilayer neural network in an off‐line session. The network is then used effectively in a real‐time for feedback guidance method. Simulation results show that this neural networks guidance method can efficiently produce an optimal feedback law in spite of relatively simple network architecture. Copyright © 2000 John Wiley & Sons, Ltd.     

Suboptimal controller design for active control of circular saw vibrations

A circular saw in operation can have an instability phenomenon which is characterized by a large amplitude and operation speed-dependent vibration. In this paper, a method of suppressing such unstable motion is presented. An optimal output feedback control scheme is adopted, and only the output variables available from sensors are used. A truncated vibrational mode model is used to formulate the saw dynamics. This model incorporates the forces due to the actuators. Since the performance of controlling the saw motion depends greatly upon spatial locations of both actuators and sensors, their locations are optimally determined with respect to a prescribed performance index representing the vibrating system energy. A two-mode simulation of a saw is made to illustrate the effectiveness of the control scheme. The results show that this proposed design method stabilizes the otherwise unstable motion of the saw and also requires a very short controlling time for vibration suppression.     

On Applications of a new method for computing optimal non-linear feedback controls

A new method of computing optimal non-linear feedbacks is used for regulating the angular momentum of spacecraft using both reaction jets and flywheels. It is shown that the optimal feedback law satisfies a system of first-order, quasi-linear, partial differential equations. The integration of these equations by the method of characteristics gives the non-linear feedback control. This control reduces the angular velocities of the space vehicle to zero by minimizing the fuel consumption. The optimal regulation, under reaction jet control alone and with the flywheels at fixed angular velocities, is considered. The special case where these velocities are zero leads readily to the known analytical solution of the feedback law, which is linear in the state although the dynamics is non-linear.     

Robust pole assignment in a specified region using output feedback

An output feedback pole assignment method is presented which seeks to achieve a robust solution where the assigned poles are as insensitive as possible to perturbations in the closed-loop system. The freedom available to obtain a robust solution is realized by assigning poles not to exact locations but within specified regions of the complex plane. The problem is formulated as a non-linear programming problem with a set of non-linear constraints of matrix-valued functions as well as a set of linear constraints. A numerical algorithm is presented in two forms and the results obtained are compared with those generated from other algorithms for a series of test examples taken from the literature.     

Design of optimal feedback controllers for minimum eigenvalue sensitivity

An iterative method for designing an optimal constant gain feedback controller for a linear system to achieve minimum eigenvalue sensitivity to parameter variations is presented. In addition to assigning eigenvalues to desired locations in the complex plane, one can also assign elements of eigenvectors by this method. This makes it possible to shape the response of the states. Examples to illustrate the effectiveness of this method are included.     

An optimal control approach to spacecraft rendezvous on elliptical orbit

In this paper, we consider a linear quadratic regulator control problem for spacecraft rendezvous in an elliptical orbit. A new spacecraft rendezvous model is established. On the basis of this model, a linear quadratic regulator control problem is formulated. A parametric Lyapunov differential equation approach is used to design a state feedback controller such that the resulting closed‐loop system is asymptotically stable, and the performance index is minimized. By an appropriate choice of the value of a parameter, an approximate state feedback controller is obtained from a solution to the periodic Lyapunov differential equation, where the periodic Lyapunov differential equation is solved on the basis of a new numerical algorithm. The spacecraft rendezvous mission under the controller obtained will be accomplished successfully. Several illustrative examples are provided to show the effectiveness of the proposed control design method. Copyright © 2014 John Wiley & Sons, Ltd.     

Characterization and selection of global optimal output feedback gains for linear time‐invariant systems

The problem of global optimal static output feedback control for linear time‐invariant systems with linear quadratic index is investigated. The contributions of this paper are two‐fold. One is to investigate the dependence of the global optimal output feedback gain on the system initial conditions. The other is to construct a globally optimal feedback under a certain output measurement structure. Copyright © 2000 John Wiley & Sons, Ltd.     

Differential dynamic programming technique for optimal control

Continuous- and discrete-time differential dynamic programming (DDP) approaches to solve general optimal control problems are described and analysed. A comparison of the two approaches shows the continuous-time approach to be more general and flexible compared with the discrete-time approach, since it is not tied to any discretization scheme. A comparison of DDP with the non-linear programming (NLP) approach is also given. Three structural control problems — a linear model of a space structure, a single degree of freedom non-linear impact absorber and a non-linear flexible beam subjected to an impulsive load — are used to numerically evaluate the continuous- and discrete-time DDP approaches. Several grid sizes are used to show that the continuous-time approach with a reasonable number of grid points is more accurate and efficient (in most cases) than the discrete-time approach. It is therefore recommended to fully develop and evaluate the technique for the optimal control of large-scale systems.     

A modified optimal load-frequency controller for interconnected power systems

This paper presents a modified optimal load-frequency controller for interconnected hydrothermal power systems. The weighting matrix of the optimal controller is sequentially modified to let the dominant energy open-loop poles or poles in a conventional optimal feedback system be located in the desired region. The poles selected to be shifted are determined using the concept of dominant energy modes rather than those located closer to the imaginary axis. Improper pole placement can be avoided and very good system frequency and tie-line power dynamic responses can be achieved by the proposed controller.     

Design of a field voltage controller for a synchronous generator using feedback linearization

The problem of non-linear field voltage controller synthesis for a synchronous generator is considered. The theory of feedback linearization of non-linear systems is applied and the existence of a linearizing feedback controller for the improved reduced-order three-dimensional model of a generator is proved. Simulation results for the proposed non-linear controller, which includes a non-linear observer and two systems based on an automatic voltage regulator and a power system stabilizer, are presented.     

Sequential rank-one/rank-two updates for quasi-newton differential dynamic programming

This communication presents computational results with a variant of differential dynamic programming. The class of methods is motivated by quasi-Newton methods and constructs quadratic approximations by using stagewise gradients. We compare two methods of updating the approximations: the first is based on the symmetric Broyden update and the second is a sequential rank-one/rank-two update. Our computational results suggest that the latter is a practical and efficient algorithm for discrete-time optimal control.     

Modelling and control of a rotary crane with a flexible joint

A dynamical model is derived for the control of a rotary crane, which has a flexible joint and makes three kinds of motion (rotation, load hoisting and boom hoisting) simultaneously. The goal is to transfer a load to a desired place in such a way that at the end of the transfer the swing of the load decays as quickly as possible. A new open-loop plus feedback control scheme is proposed.