Linear Feedback Control Systems Research Articles

Output Feedback Controllers for Linear Systems With Delayed Perturbations

This paper presents an approach to the synthesis of robust memoryless H∞ controller for the uncertain state delayed system where full state is not available and output is used for feedback control. The controller guarantees that the closed-loop system is stable and achieves a prescribed level of disturbance attenuation for all admissible parameter uncertainties. A numerical example is given to illustrate the results.

Control of continuous-time systems with discrete jumps

AbstractIn this paper, the design of output feedback controllers for linear systems under sampled measurements is investigated. The performance we use is the worst-case gain from disturbances to the controlled output, which comprises both a continuous-time and a discretetime signal to be controlled. Control problems in both the finite and infinite horizonare addressed. Necessary and sufficient conditions for the existence of a suitable sampled-data output feedback controller are given in terms of two Riccati differential equations with finite discrete jumps. A numerical example is given to show the potential of the proposed technique.

Parametric model development and quantitative feedback design for automotive torque converter bypass clutch control

Customer demands for large high-performance vehicles in the face of increasingly stringent fuel economy targets have led automobile makers to seek innovative ways of meeting these requirements, especially in the North American market. One of the most critical elements in automotive powertrain systems is the torque converter bypass clutch which is now in almost universal use in automatic transmission-equipped vehicles. The clutch is used to bypass the fluid coupling of the torque converter during steady state conditions (i.e. highway cruise), thus eliminating a 15–20 percent loss of efficiency. Generally, the clutch is disengaged during dynamic events (e.g. vehicle launch, braking and gear changes) when the torque converter functionality is required. Further gains in fuel economy can now be realized through modulation of the bypass clutch in some situations where it had been considered necessary that the clutch be disengaged (i.e. during gear changes). In this case, the clutch is operated in a state of continuous slip; the transmitted torque is controlled through modulation of the clutch apply pressure. The dynamics of this system, however, exhibit considerable complexity. Characterization of this behaviour has been of considerable interest among automobile makers in recent years. However, investigation of the non-linear dynamics of this system is beyond the scope of this paper; a more in-depth treatment generally requires bifurcation analysis and/or exhaustive simulation studies. The focus in this paper is upon the detailed development of a linear parametric differential equation model and the design of a linear robust feedback control system. Parametric uncertainty is included to capture the effects of variations in system damping, bulk modulus, coefficient of friction and constants of linearization. Based upon a specific operating point, a linear robust controller is developed using the quantitative feedback theory (QFT) technique. The QFT methodology is aimed at designing feedback controllers so that pointwise frequency response specifications on closed-loop tracking and disturbance rejection are met in spite of large parametric plant uncertainty. The local stability and performance of the non-linear feedback system are subsequently verified by simulation. Since the feedback design is based upon a linear parametric model, no specific guarantees can be made as to the performance of the non-linear closed-loop system, although the results are found to be satisfactory in this case.

Control system analysis of an annular prime mover

The thermoacoustic prime mover in an annular resonator is analyzed as a linear feedback control system. Using the techniques of control system analysis, a closed-form solution for the acoustic gain of a constriction with linear temperature gradient in an annular resonator is developed. For a large enough gradient in the constriction, the acoustic gain of the system could become infinite, at which time the system goes into unstable positive feedback creating a thermoacoustic prime mover. An analysis of an existing annular resonator system is made and a comparison to more standard thermoacoustic numerical analysis is presented. [Work supported by the Office of Naval Research and the American Society for Engineering Education.]

Performance assessment of multivariable feedback controllers

Procedures are developed for assessing the performance of multi-input multi-output (MIMO) linear feedback control system. The predicted performance of the MIMO minimum variance controller is used as a lower bound against which current performance is assessed. To use the procedures, one needs only a set of representative data on the controlled variables collected under the current feedback control scheme, and an estimate of the process interactor matrix characterizing the dead-time structure of the process. A non-parametric correlation test allows for a quick check on the closeness of the performance of the current feedback control system to the theoretical lower bound. A quantitative analysis then provides an estimate of the theoretical lower bound on the variance-covariance matrix of the controlled variables, and on the quadratic performance objective of the minimum-variance controller. A measure of controller performance based on the ratio of the current value of the quadratic objective to this thoeretical lower limit can then be monitored on-line. The purpose of these methods is to allow engineers to assess the current performance of the existing control systems, to monitor how this performance changes over time, to see which variables are well controlled and which are not, and to help in deciding whether or not there is sufficient incentive to reidentify the process model and/or redesign the controller.

Control of wing rock motion using adaptive feedback linearization

The theory of adaptive control of feedback linearizable systems is applied to designing the control of wing rock motion with the extension of the technique to include tracking. The adaptation law is designed to adjust the aerodynamic parameters in the model. Precise tracking and maximum performance can be achieved if sufficient rolling moment derivative because of lateral control components are available. Case studies and simulation are carried out to illustrate the results.

On relative stability of linear stationary feedback control systems with time delay

On relative stability of linear stationary feedback control systems with time delay

A multilayer recurrent neural network for on-line synthesis of minimum-norm linear feedback control systems via pole assignment

A multilayer recurrent neural network is proposed for on-line synthesis of minimum-norm linear feedback control systems through pole assignment. The proposed neural network approach uses a four-layer recurrent neural network for the on-line computation of feedback gain matrices with the minimum Frobenius norm and desired closed-loop poles. The proposed recurrent neural network is shown to be capable of synthesizing minimum-norm linear feedback control systems in real time. The operating characteristics of the recurrent neural network and feedback control systems are demonstrated by use of an illustrative example.

Dynamics and stability state-space of a controlled inverted pendulum

A study is conducted into the dynamics of an inverted pendulum under the regulation of a linear-feedback control system. Analysis of the resulting dynamic equations indicates the presence of stable regions within the four-dimensional state space of the system, which are confirmed through simulation. System trajectories are examined in this state space, and complex, apparently chaotic behaviour is observed outside the stable regions of the state space. An algorithm is developed to map the boundaries of the stable state-space hyper-volume; this algorithm is broadly applicable to the study of other systems. Results are presented, generated by executing the algorithm on the inverted pendulum dynamics. These comprise a series of three-dimensional plots, outlining the four-dimen-sional stability volume. The boundaries of this volume are seen to exhibit a jagged, possibly fractal, morphology.

Nonlinear predictive control of feedback linearizable systems and flight control system design

The question of output trajectory control of input-output feedback linearizable nonlinear dynamic systems using state variable feedback is considered. For the derivation of the predictive control law, a vector function s is chosen as a linear combination of the tracking error, its higher order derivatives, and the integral of the tracking error. The control law is obtained by the minimization of a quadratic function of the predicted value of s and the control input. It is shown that in the closed-loop system the trajectories are uniformly ultimately bounded in the presence of uncertainty in the system parameters. Based on these results, a flight control system for the trajectory control of the output variables pitch, sideslip, and roll angles (0, /3, ) using aileron, rudder, and elevator control is presented. Simulation results are obtained to show that precise simultaneous longitudinal and lateral maneuvers can be performed in spite of the uncertainty in the aerodynamic parameters.

Linearly pulse-width modulated block pulse functions and their application to linear SISO feedback control system identification

A modification to conventional block pulse functions (BPF) is proposed by describing a set of linearly pulse-width modulated block pulse functions (LPWM-BPF) that has been utilised to develop a generalised convolution matrix of operational nature. This matrix is used to determine convolution of time-varying functions and is also employed to solve linear feedback system identification problem. Also, a recursive technique for solving the identification problem in the conventional BPF domain has been derived. Numerical examples are treated to establish the validity of the proposal. The representational error analysis has also been carried out for the LPWM-BPF to show that this kind of BPF set introduces less error than the conventional equal-width BPF.

Feedback control of linear systems by multirate pulse-width modulation

In this note, sampled-data methods for modeling and control design of pulse-width modulated systems are examined. Both single-rate and multirate approaches are considered. Local controllability and observability analyses of the sampled-data models are performed, and feedback linearizability conditions for the sampled-data models are derived. Multirate nonlinear state and output feedback designs are presented that permit asymptotic regulation or tracking. >

Time-domain Shaping of Linear Systems via Dual Formulation

Results are presented which generalize the known solution to the problem of minimizing the l1 norm of the error transfer function. The optimization framework is a dual formulation employing the theory of convex functionals applied to the problem of designing optimal compensators for linear feedback control systems. A wide range of cost functions can be optimized using this theory and it allows a rather general treatment of the topic of time-domain shaping of linear systems. The results obtained can be applied to the mathematically equivalent problems of optimal disturbance rejection and optimal tracking.

Genetic design of linear multivariable feedback control systems using eigenstructure assignment

In the design of linear multivariable feedback control systems, performance specifications can frequently be met by assigning appropriate closed-loop eigenstructure. However, although various powerful algorithms for the solution of this problem of eigenstructure assignment have been proposed over the years, no universally applicable and computationally stable methodology has yet emerged. These difficulties motivate the use of genetic algorithms in the solution of the fundamental problem of eigenstructure assignment.

Multipurpose controller design using parametric optimization with dynamic constraints

A systematic and effective design procedure for a class of linear multivariable feedback control systems is proposed For simultaneously achieving robust tracking of the prescribed input signal via the Q-parametrization approach and internal model principle, the prescribed transient specifications by choosing an appropriate performance criterion, robust stabilization, and sensitivity minimization. The problem can best be cast in the form of an optimization problem with dynamic constraints. A numerical method is formulated to solve the optimization problem with dynamic constraints using a non-linear programming method. A numerical example is provided to illustrate the main results of the paper.

Applications of Routh-Hurwitz criterion to evaluation of phase margin and phase lead compensation

A method, based on a transformation G(s) = ½[aG(s) + G(s)−1], is proposed for applying the Routh-Hurwitz criterion to determine the phase margin of a linear time-invariant feedback control system, and to design phase-lead compensation. A numerical example is also given to illustrate the method.

Estimation of Nonstationary Armax Models Based on the Hannan-Rissanen Method

We consider in this paper the estimation problems for both orders and coefficients of linear feedback control systems, described by ARMAX models. The estimation algorithms are inspired by the Hannan-Rissanen method used for the estimation of stationary ARMA models, while the convergence analyses are based on limit theorems for both double array martingales and nonnegative supermartingales, and on techniques of stochastic Lyapunov functions. Traditionally used assumptions, such as the strictly positive real condition and the requirement of known upper bounds for true orders, are not imposed here.

Quantizer effects on steady-state error specifications of digital feedback control systems

Suppose that for a linear, time-invariant, single-input single-output (SISO) plant a digital dynamic controller has been designed so that the corresponding discrete-time linear feedback control system will satisfy certain specified steady-state error criteria (e.g., the output v(kT) will follow the input r(kT) within a certain given error bound). Corresponding to this idealized linear system, the digital implementation of the dynamic controller will result in a nonlinear feedback control system. To what extent the resulting nonlinear digital feedback control system satisfies the steady-state-error criteria of the original idealized linear prototype feedback control system is investigated. Digital controller implementations which use fixed-point arithmetic or floating-point arithmetic are considered. Designs which are well scaled are assumed, i.e., overflow nonlinearities in digital controllers are not addressed. >

Stability Limits of Linear Feedback Control Systems

A new method is presented for determining the stability limits of open loop gains in linear feedback controlled systems. The method does not require finding the roots of multiple polynomials in k, the loop gain, as in the case of the commonly used algebraic Routh array technique. Instead, the roots of a single polynomial in s (the Laplace variable) of order at least one less than that of the system are evaluated to directly determine where the root locus crosses the imaginary axis.

Least-squares identification for ARMAX models without the positive real condition

Recursive identification problems of linear stochastic feedback control systems described by ARMAX models are studied without imposing the strictly positive real condition on the noise model. An increasing lag least squares is used in the first step to estimate the noise process, while the parameter estimate is formed in the second step by an extended least squares. In the algorithm, there is an increase in computational cost in comparison to traditional algorithms. However the results of this note do not need any a priori information or conditions on the noise model except that of stability, and they are applicable to the identification of general feedback control systems. >