Optimal controls for an advanced turbofan engine

A mathematical model for the steering control of an automobile is described. The structure of the model derives from linear optimal discrete time preview control theory but it is non-linear. Its parameter values are obtained by heuristic methods, using insight gained from the linear optimal control theory. The driver model is joined to a vehicle dynamics model and the path tracking performance is demonstrated, using moderate manoeuvring and racing speeds. The model is shown to be capable of excellent path following and to be robust against changes in the vehicle dynamics. Application to the simulation of manoeuvres specified by an ideal vehicle path and further development of the model to formalise the derivation of its parameter values and to put it to other uses are discussed.

A EROSPACE engineering applications greatly stimulated the development of optimal control theory during the 1950s and 1960s, where the objective was to drive the system states in such a way that some defined cost was minimized. This turned out to have very useful applications in the design of regulators (where some steady state is to be maintained) and in tracking control strategies (where some predetermined state trajectory is to be followed). Among such applications was the problem of optimal flight trajectories for aircraft and space vehicles. Linear optimal control theory in particular has been very well documented and widely applied, where the plant that is controlled is assumed linear and the feedback controller is constrained to be linear with respect to its input. However, the availability of powerful low-cost microprocessors has spurred great advantages in the theory and applications of nonlinear control. The competitive era of rapid technological change, particularly in aerospace exploration, now demands stringent accuracy and cost requirements in nonlinear control systems. This has motivated the rapid development of nonlinear control theory for application to challenging, complex, dynamical real-world problems, particularly those that bear major practical significance in aerospace, marine, and defense industries. Infinite-time horizon nonlinear optimal control (ITHNOC) presents a viable option for synthesizing stabilizing controllers for nonlinear systems by making a state-input tradeoff, where the objective is to minimize the cost given by a performance index. The original theory of nonlinear optimal control dates from the 1960s. Various theoretical and practical aspects of the problem have been addressed in the literature over the decades since. In particular, the continuous-time nonlinear deterministic optimal control problem associated with autonomous (time-invariant) nonlinear regulator systems that are affine (linear) in the controls has been studied by many authors. The long-established theory of optimal control offers quite mature and well-documented techniques for solving this control-affine nonlinear optimization problem, based on dynamic programming or calculus of variations, but their application is generally a very tedious task. Bellman’s dynamic programming approach reduces to solving a nonlinear first-order partial differential equation (PDE), expressed by the Hamilton–Jacobi–Bellman (HJB) equation. The solution to the HJB equation gives the optimal performance/cost value (or storage) function and determines an optimal control in feedback form under some smoothness assumptions. Alternatively, in the classical calculus of variations, optimal control problems can be characterized locally in terms of the Hamiltonian dynamics arising from Pontryagin’s minimum principle. These are the characteristic equations of the HJB PDE, which result in a nonlinear, constrained two-point boundary value problem (TPBVP) that, in general, can only be solved by successive approximation of the optimal control input using iterative numerical techniques for each set of initial conditions. Numerically, even though the nonlinear TPBVP is somewhat easier to solve than the HJBPDE, control signals can only be determined offline and are thus best suited for feedforward control of plants for which the state trajectories are known a priori. Therefore, contrary to the dynamic programming approach, the resultant control law is not generally in feedback form. Open-loop control, however, is sensitive to random disturbances and requires that the initial state be on the optimal trajectory. In contrast, nonlinear optimal feedback has inherent robustness properties (inherent in the sense that it is obtained by ignoring uncertainty and disturbances). The potential difficulty with the HJB approach is that no efficient algorithm is available to solve the PDE when it is nonlinear and the problem dimension is high, making it impossible to derive exact expressions for optimal controls for most nontrivial problems of interest. The optimal can only be computed in special cases, such as linear dynamics and quadratic cost, or very low-dimensional systems. In particular, if the plant is linear time invariant (LTI) and the (infinite-time) performance index is quadratic, then the corresponding HJB equation for this infamous linear-quadratic regulator (LQR) problem reduces to an algebraic Riccati equation (ARE). Contrary to the well-developed and widely applied theory and computational tools for theRiccati equation (for example, see [1]), theHJB equation is difficult, if not impossible, to solve for most practical applications. The exact solution for the optimal control policies is very complex

The development of on-ramp metering control strategies for a freeway can be aided by the well-developed theory of linear optimal control. However, an adequate model of an appropriate form must be available. Recently developed "aggregate variable" models appear to be best suited to this purpose. In these models the freeway is sectioned into lengths of about <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\frac{1}{2}</tex> mi and traffic is described in terms of the aggregate variables section speed, section density, and volume. The model is used to formulate a linear regulator problem with quadratic cost for the purpose of determining a feedback control rule which returns the freeway to nominal conditions after a disturbance. The solution of the regulator problem involves considerable computation difficulty when the order of the system is large. In a large class of applications the optimal control law has a local structure which can be exploited in a computationally efficient manner to yield a suboptimal control with little loss of system performance. A new technique, the augmentation method, is presented for constructing a suboptimal control from a set of optimal controls for low-order sub-systems. New results are presented which provide a rational and elegant measure for the additional cost associated with an arbitrary stabilizing suboptimal control. Additional results pertain especially to the augmentation procedure. The models and suboptimal control scheme are applied to a specific detailed design for an 8-mi segment of the San Diego Freeway in Los Angeles. Simulations are shown illustrating the effectiveness of such traffic-responsive controls in clearing up congestion due to traffic incidents.

The design method of a quantitative robust optimal control system is developed. Wiener-Hopf linear quadratic optimal control and quantitative feedback theory are applied to achieve the optimal system control. In this design, two frequency-dependent weighting functions are shaped and two-degree-of-freedom compensators are designed to achieve the robust optimal control.

The use of an adaptive optimal control algorithm for two real-time control applications, an optimal excitation control of a synchronous generator (OEC) and a power system stabilizer (PSS), is described. Experimental studies on a physical model of a power system showed that the proposed OEC and PSS can track the controlled system by parameter identification at different operating conditions. The proposed algorithm is based on linear optimal control theory. A special fifth-order discrete Riccati equation is solved in each sampling period. The proposed OEC and PSS can always guarantee that in a closed loop the controlled system will be stable based on the identified parameters. As the actual output of the controlled system and control are directly used in the controller, no observer is required. The proposed OEC and PSS can track the controlled system very rapidly. A number of tests have been performed, and all show satisfactory results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The development of on ramp metering control strategies for a freeway can be aided by the well-developed theory of linear optimal control. The recently developed variable model appears to be best suited for this purpose. In this model the freeway is partitioned into lengths of about 1/2 mile and traffic is described in terms of the aggregate variables section speed, section density, and volume. This model and linear optimal control theory are used to developed an on ramp metering strategy which is designed to maintain freeway conditions near nominal conditions in the presence of disturbances. An illustrative design for a segment of the San Diego freeway in Los Angeles is provided. The companion problem of processing freeway sensor data to provide current estimates of the aggregate variables can be approached through standard techniques in estimation theory. Methods of attacking this portion of the freeway surveillance problem are presented.

A fuzzy logic adaptive Kalman filtering methodology was developed for the automatic control of an irrigation canal system under unknown disturbances (water withdrawals) acting in the canal. Using a linearized finite difference model of open channel flow, the canal operation problem was formulated as an optimal control problem and an algorithm for gate opening in the presence of arbitrary external disturbances (changes in flow rates) was derived. Based on the linear optimal control theory, the linear quadratic regulator (LQR), assuming all the state variables (flow depths and flow rates) were available, was designed to generate control input (optimal gate opening). As it was expensive to measure all the state variables (flow rates and flow depths) in a canal system, a fuzzy logic adaptive Kalman filter and traditional Kalman filter were designed to estimate the values for the state variables that were not measured but were needed in the feedback loop. The performances of the state estimators designed using the fuzzy logic adaptive Kalman filter methodology and the traditional Kalman filtering technique were compared with the results obtained using the LQR (target loop function). The results of the present study indicated that the performance of the fuzzy logic adaptive Kalman filter was far superior to the performance of the observer design based upon the traditional Kalman filter approach. The obvious advantages of the fuzzy logic adaptive Kalman filter were the prevention of filter divergence and ease of implementation. As the fuzzy logic adaptive Kalman filter requires smaller number of state variables for the acceptable accuracy therefore, it would need less computational effort in the control of irrigation canals. Copyright © 2009 John Wiley &amp; Sons, Ltd.

This paper deals with a theoretical study of the active control of purely axial 1D instabilities in multi-stage axial-flow compressors. The paper briefly considers the development of a suitable surge model and continues with the derivation of a controller using linear optimal control theory. The controller is specifically designed to suppress the instabilities predicted by the linearised form of the surge model. The control technique involves a bleed being dynamically varied in response to fluctuations of variables. It is found that a stabilising optimal controller can always be obtained. The second part of the paper presents a non-linear simulation of the surge prediction model with the linear controller, using a 4th order Runge-Kutta scheme. This demonstrates how some non-linearities in the flow model can affect the controller operation. It is also shown that, the amplitudes and the frequency response required of the actuator to retain stability can exceed the practical limitations.Copyright © 1993 by ASME

A recurrent dynamic neural network (RDNN) based on Hopfield model is applied to linear quadratic regulator controller of an irrigation canal. The Saint-Venant equations of open-channel flow are linearized using the Taylor series and a finite difference approximation of the original nonlinear, partial differential equations. Using the linear optimal control theory, a Linear Quadratic Regulator (LQR) controller is developed for an irrigation canal with a single pool. A Hopfield recurrent dynamic neural network is applied to solution of Riccati equation for calculating the LQR control gains. The main advantage of Hopfield neural network model is its robustness and flexibility when dealing with unknown perturbations and parameters. The results of this study shows that an RDNN based LQR controller (RDNN/LQR) for irrigation canals offers an alternative to traditional LQR controller when dealing with the disturbances in the irrigation canals.

A physiologically plausible model for control of human postural balance, combining optimal state estimation and control, is proposed. The linear dynamics of three sensory modalities are modeled: joint proprioception, vestibular organs in the inner ear, and vision. These sensors are mated with a two degree of freedom model of planar body dynamics. Linear quadratic optimal control theory is used to design the state feedback and estimation gains. Free parameters for the overall system set the form of the control objective and the sensory precision of the sensors. The model predicts statistical properties of human sway in terms of covariance of ankle and hip motion. These predictions are compared with human responses to alterations in sensory conditions. For a single set of parameters, the model successfully reproduces the general nature of postural motion as a function of sensory environment. Sensitivity studies demonstrate that the model is robust to parameter variations, consistent with observed behavior. A physiological structure is proposed for learning and implementing the necessary computations. The model may be useful as a diagnostic tool for detecting contributing factors to poor balance.

Multiple coordinated robot arms are modeled by considering the arms as closed kinematic chains and as a force-constrained mechanical system working on the same object simultaneously. In both formulations, a novel dynamic control method is discussed. It is based on feedback linearization and simultaneous output decoupling technique. By applying a nonlinear feedback and a nonlinear coordinate transformation, the complicated model of the multiple robot arms in either formulation is converted into a linear and output decoupled system. The linear system control theory and optimal control theory are used to design robust controllers in the task space. The first formulation has the advantage of automatically handling the coordination and load distribution among the robot arms. In the second formulation, it was found that by choosing a general output equation it became possible simultaneously to superimpose the position and velocity error feedback with the force-torque error feedback in the task space. >

Dynamic Vibration Absorbers (DVAs) have been used to attenuate vibrations in various types of mechanical Systems. In its simplest form, a DVA is formed by an association of passive elements (inertia, stiffness and damping). The values of these parameters are selected so as to tune the DVA to a given value of the excitation frequency, assumed to be fixed. As a result, the attenuation capability of a passive DVA significantly decreases as the excitation frequency deviates from the nominal tuning frequency. To avoid this drawback, active and adaptive DVAs have been extensively studied lately, in an attempt to achieve larger effective bandwidths and self tunning capability. Active DVAs contain, besides the passive elements, an actuator wich applies a control force determined by an appropriate control law. Adaptive DVAs are understood as those constructed in such a way that the values of their physical parameters can be adjusted according to well-defined laws. In this work, some configurations of active and adaptive DVAs are assessed, namely: a) an active DVA using as feedback signal the time-delayed displacement response of the reactive mass (delayed resonator); b) a novel active DVA using a control law in wich the control force is expressed as a linear combination of the displacement, velocity and acceleration of the reactive mass, relative to the primary mass; c) an active DVA exploring linear quadratic optimal control theory; d) a vibrating string-type adaptive DVA; e) a pendulum-type adaptive DVA; f) an adaptive DVA formed from a beam with piezoelectric patches. For each configuration the theoretical foundations are first developed. Then, numerical applications are presented to assess their main features and performance.

Rapid advances in information processing, communication and sensing technologies have enabled more and more devices to be provided with embedded processors, networking capabilities and sensors. For the field of estimation and control, it is now possible to consider an architecture in which many simple components communicate and cooperate to achieve a joint team goal. This distributed (or networked) architecture promises much in terms of performance, reliability and simplicity of design; however, at the same time, it requires extending the traditional theories of control, communication and computation and, in fact, looking at a unified picture of the three fields. A systematic theory of how to design distributed systems is currently lacking. This dissertation takes the first steps towards understanding the effects of imperfect information flow in distributed systems from an estimation and control perspective and coming up with new design principles to counter these effects. Designing networked systems is difficult because such systems challenge two basic assumptions of traditional control theory - presence of a central node with access to all the information about the system and perfect transmission of information among components. We formulate and solve many problems that deal with the removal of one, or both, of these assumptions. The chief idea explored in this dissertation is the joint design of information flow and the control law. While traditional control design has concentrated on calculating the optimal control input by assuming a particular information flow between the components, our approach seeks to synthesize the optimal information flow along with the optimal control law that satisfies the constraints of the information flow. Thus besides the question of 'What should an agent do?', the questions of 'Whom should an agent talk to?', 'What should an agent communicate?', 'When should an agent communicate?' and so on also have to be answered. The design of the information flow represents an important degree of freedom available to the system designer that has hitherto largely been ignored. As we demonstrate in the dissertation, the joint design of information flow and the optimal control input satisfying the constraints of that information flow yields large improvements in performance over simply trying to fit traditional design theories on distributed systems. We begin by formulating a distributed control problem in which many agents in a formation need to cooperate to minimize a joint cost function. We provide numerical algorithms to synthesize the optimal constrained control law that involve solving linear equations only and hence are free from numerical issues plaguing the other approaches proposed in the literature. We then provide and analyze a model to understand the issue of designing the topology according to which the agents interact. The results are very surprising since there are cases when allowing communication to happen between two agents may, in fact, be detrimental to the performance. We then move on to consider the effects of communication channels on control performance. To counter such effects, we propose the idea of encoding information for the purpose of estimation and control prior to transmission. Although information theoretic techniques are not possible in our problem, we are able to solve for a recursive yet optimal encoder / decoder structure in many cases. This information flow design oriented approach has unique advantages such as being optimal for any packet drop pattern, being able to include the effect of known but random delays easily, letting us escape the limits set by reliability for transmission of data across a network by using intermediate nodes as 'repeaters' similar to a digital communication network and so on. We finally take a look at combining the effects of multiple sources of information and communication channels on estimation and control. We look at a distributed estimation problem in which, at every time step, only a subset out of many sensors can transmit information to the estimator. This is also a representative resource allocation problem. We propose the idea of stochastic communication patterns that allows us to include the effects of communication channels explicitly during system design. Thus, instead of tree-search based algorithms proposed in the literature, we provide stochastic scheduling algorithms that can take into account the random packet drop effect of the channels. We also consider a distributed control problem with switching topologies and solve for the optimal controller. The tools that we develop are applicable to many other scenarios and we demonstrate some of them in the dissertation. Along the way, we look at many other related problems in the dissertation. As an example, we provide initial results about the issue of robustness of a distributed system design to a malfunctioning agent. This notion is currently lacking in the control and estimation community, but has to be a part of any effective theory for designing networked or distributed systems.

A stability and control augmentation system (SCAS) was designed based on a set of comprehensive performance criteria. Linear optimal control theory was applied to determine appropriate feedback gains for the stability augmentation system (SAS). The helicopter was represented by six-degree-of-freedom rigid body equations of motion and constant factors were used as weightings for state and control variables. The ratio of these factors was employed as a parameter for SAS analysis and values of the feedback gains were selected on this basis to satisfy three of the performance criteria for full and partial state feedback systems. A least squares design method was then applied to determine control augmentation system (CAS) cross feed gains to satisfy the remaining seven performance criteria. The SCAS gains were then evaluated by nine degree-of-freedom equations which include flapping motion and conclusions drawn concerning the necessity of including the pitch/regressing and roll/regressing modes in SCAS analyses.

A class of linear regulator problems, referred to as tracking problems, impose the requirements of zero steady state error, elimination of disturbance effects and immunity to system parameter changes. A controller design approach for this problem, using linear optimal control theory, is presented, and its advantages are outlined by comparison with previous modern and classical approaches.