LQG Optimal Control Research Articles

An exact solution to continuous-time mixed H/sub 2//H/sub /spl infin/#x221E;/ control problems

Multiobjective control problems have been the object of much attention in the past few years, since they allow for handling multiple, perhaps conflicting, performance specifications and model uncertainty. One of the earliest multiobjective problems is the mixed H/sub 2//H/sub /spl infin// control problem, which can be motivated as a nominal LQG optimal control problem subject to robust stability constraints. This problem has proven to be surprisingly difficult to solve, and at this time no closed-form solutions are available. Moreover, it has been shown that except in some trivial cases, the optimal controller is infinite-dimensional. In this paper, we propose a solution to general continuous-time mixed H/sub 2//H/sub /spl infin// problems, based upon constructing a family of approximating problems, obtained by solving an equivalent discrete-time problem. Each of these approximations can be solved efficiently, and the resulting controllers converge strongly in the H/sub 2/ topology to the optimal solution.

LQG optimal control of constrained discrete time systems

In this paper, we study the discrete time LQG optimal control problem, with convex quadratic con straints. We show that the Separation Theorem does not hold. However, a generalization of this result which we call the Quasi-Separation Theorem applies. The optimal control is obtained by solving a convex, finite-dimensional optimization problem that is equivalent to solving a sequence of unconstrained LQG control problems.

Minimax LQG optimal control of acoustic noise in a duct

This paper presents a new approach to the design of robust active noise controllers for an acoustic duct. This involves using a recently developed minimax LQG design methodology. The controller is designed to achieve robustness against variations in the resonant frequency of the loudspeaker.

Multiobjective optimal structural control of the Notre Dame building model benchmark

Reduced-order, multiobjective optimal controllers are developed for the Notre Dame structural control building model benchmark. Standard H 2 /LQG optimal control excels at noise and disturbance rejection, but may have difficulty with actuator saturation and plant uncertainty. The benchmark problem is adapted to a multiobjective optimal control framework, using l 1 and H∞ constraints to improve controller performance, especially attempting to reduce peak responses, avoid saturation, and improve robustness to unmodelled dynamics. The tradeoffs between H 2 performance, output peak magnitudes, and robust stability are examined. Several optimal controllers and their performance on the benchmark are given.

Track-Keeping Controller for a Precision Manoeuvring Autopilot

An autopilot for precise manoeuvring at low speed requires special attention to a number of factors including the effects of wind and current. The control system for track-keeping described here is based on the water-relative ship dynamics modelling and is characterised by the arbitrary tracking capability and the wind/current compensation within the LQG Optimal Control design.

On Plant and LQG Controller Continuity Questions

Using the dual Youla parametrizations of controller-based coprime factor plant perturbations and plant-based coprime factor controller perturbations, we study the LQG plant-controller continuity question. Indeed, we show that it is possible to calculate a new optimal LQG controller from a previous one when the plant is slightly changed, and to quantify the change in the controller as a function of the change in the plant. In addition, we compute the degradation in the achieved LQG cost when the LQG controller is computed on the basis of a perturbation of the real plant. As a by-product, we characterize the set of all plants that have the same optimal LQG controller.

An exact solution to general four-block discrete-time mixed ℋ/sub 2//ℋ/sub ∞/ problems via convex optimization

The mixed /spl Hscr//sub 2//spl Hscr//sub /spl infin// control problem can be motivated as a nominal LQG optimal control problem subject to robust stability constraints, expressed in the form of an /spl Hscr//sub /spl infin// norm bound. This paper contains a solution to a general four-block mixed /spl Hscr//sub 2///spl Hscr//sub /spl infin// problem, based upon constructing a family of approximating problems. Each one of these problems consists of a finite-dimensional convex optimization and an unconstrained standard /spl Hscr//sub /spl infin// problem. The set of solutions is such that in the limit the performance of the optimal controller is recovered, allowing one to establish the existence of an optimal solution. Although the optimal controller is not necessarily finite-dimensional, it is shown that a performance arbitrarily close to the optimal can be achieved with rational controllers. Moreover, the computation of a controller yielding a performance /spl epsiv/-away from optimal requires the solution of a single optimization problem, a task that can be accomplished in polynomial time.

A quasi-separation theorem for LQG optimal control with IQ constraints

We consider the deterministic, the full observation and the partial observation LQG optimal control problems with finitely many IQ (integral quadratic!) constraints, and show that Wohnam's famous Separation Theorem for stochastic control has a generalization to this case. Although the problems of filtering and control are not independent, we show that the interdependence of these two problems is so superficial that in effect, they are problems which can be treated separately. It is in this context that the label Quasi-Separation Theorem is to be understood. We conclude with a discussion of computation issues and show how gradient-type optimization algorithms can be used to solve these problems. In this way, a systematic computation algorithm is derived.

On the Control of a Building Model Subject to Seismic Excitation

Abstract The control problem of a building model, subject to seismic excitation, is considered in this paper. First, a feed-forward compensator is designed to compensate the ground acceleration; secondly, an LQG optimal feedback controller is added to the control scheme to improve the control performances. In order to reduce the complexity of the resulting controller, the computations design procedure is carried out with respect to an approximate model of the building, obtained by eliminating some of the state variables from a balanced state space realization of the original model. Simulation runs confirm the effectiveness of the proposed control structures.

Dynamic Positioning of Floating Vessels Postoptimal Analysis

Abstract A procedure for the postoptimal analysis of dynamic positioning control system of floating vessels is proposed. The control system design is based on the optimal constrained covariance control (OC 3 ). Using the OC 3 technique, the disadvantages of the classical LQG optimal control technique are avoided. The presented numerical example illustrates the properties of the new approach.

Steady-state optimal discrete-time control of first-order systems with actuator noise variance linearly related to actuator signal variance

In this paper, the steady-state discrete-time optimal LQG control problem is considered when the noise variance of an actuator depends linearly on the variance of the actuator signal. A first-order system is analyzed in the case of noise-free and noisy output measurements. In the standard LQG problem the actuator noise variance is assumed to be known and constant before the design of the feedback gain. Consequently, the standard LQG design can lead to the deterioration of the system performance in the considered case.

Investigations on the Stochastically Optimized PID Controller for a Linear Quarter-Car Road Vehicle Model

SUMMARY An active suspension for a linear quarter - car road vehicle model using a PID structured controller is discussed here. The PID parameters are not optimized using the conventional methods but are based on stochastic optimal control methods. The main idea is to obtain a full feedback optimal LQG controller and then to constrain this controller to fit a PID structure using constrained feedback techniques. The overall performance of this active suspension is evaluated here.

Linearly Constrained LQ and LQG Optimal Control

It has recently been shown that the logarithmic barrier method for solving finite-dimensional, linearly constrained quadratic optimization problems can be extended to an infinite-dimensional setting with complexity estimates similar to the finite dimensional case. As a consequence, an efficient computational method for solving the linearly constrained LQ control problem is now available. In this paper, we solve the linearly constrained LQG control problem by generalizing the Separation Theorem. We show how the logarithmic barrier method can be used to determine the optimal control for the constrained LQG problem.

Optimal LQG and GPC Controllers: Covariance Characterization, Structured Models and Chandrasekhar Equations

A unified approach to the MV, LQG and GPC control problems, based on structured state-space models, is presented. Attention is paid to the computational efficiency of the methods proposed. Chandrasekhar-type equations for receding-horizon control problems are presented and the Box-Jenkins model is utilized to increase versatility and to improve computational efficiency. An integral action of the controller is attained without resorting to nonrealistic nonstationary disturbance models. Formulae for calculating the evolution of output and control variances are also derived.

On the relative entropy of discrete-time Markov processes with given end-point densities

Given a Markov process x(k) defined over a finite interval I=[0,N], I/spl sub/Z we construct a process x*(k) with the same initial density as x, but a different end-point density, which minimizes the relative entropy of x and x*. It is shown that x* is a Markov process in the same reciprocal class as x. In the Gaussian case, the minimum relative entropy problem is related to a minimum energy LQG optimal control problem.

Continuity properties of LQG optimal controllers

It is shown that the LQG optimal controller is a continuous function of the plant. The result is proved for a class of plants which contains the class of strictly proper finite-dimensional plants. The topology employed is the one generated by convergence of the closed-loop transfer functions in an induced L ∞ sense. This topology is slightly stronger than the usual H 2 gap metric convergence on transfer functions.

Two-degree-of-freedom linear quadratic Gaussian predictive control

A two-degree-of-freedom LQGPC optimal control law is derived, which has properties in common with both the LQG and GPC control laws. The stability and robustness properties are the same as for an LQG optimal controller, but the cost of future predictive error and control action is dealt with in the same manner as for the GPC control law. An advantage of the approach is that true predictive action is possible, so the control at time t will optimally use the future reference-trajectory knowledge. The feedback controller to be implemented is time-invariant, but the future predicted control action is obtained from the polynomial equivalent of a time-varying solution, which is analogous to the solution of the finite-time deterministic LQ optimal control problem. The results can be presented in a form similar to that employed in GPC algorithms, suggesting an immediate way of achieving input and output constraints.

Active control of vibration of a magnetic levitation platform

A study on active control of vibration isolation of a magnetic levitation platform used for the calibration of precision inertia instruments is presented in this paper. Nonlinear and linearized dynamic models of the system in state‐space were deduced by using the theory of an electromagnetic field. The stability of both the open‐ and closed‐loop system was investigated in the cases of different configuration of the current coils. A LQG optimal control strategy was employed in the control syntheses of the magnetic levitation system, since the system input perturbance is a stochastic excition acting on the fixed base from the ground. The transfer functions from the exciting current or the control current or the base displacement to the vibratory displacement of the levitation platform were verified experimentally. Open‐ and closed‐loop vibration responses were calculated. The results show that compared to the base displacement, the vibration of the magnetic levitation platform can be attenuated more than 20...

Relating H2 and H∞ bounds for finite-dimensional systems

For a linear time invariant system, the infinity-norm of the transfer function can be used as a measure of the gain of the system. This notion of system gain is ideally suited to the frequency domain design techniques such as H ∞ optimal control. Another measure of the gain of a system is the H 2 norm, which is often associated with the LQG optimal control problem. The only known connection between these two norms is that, for discrete time transfer functions, the H 2 norm is bounded by the H ∞ norm. It is shown in this paper that, given precise or certain partial knowledge of the poles of the transfer function, it is possible to obtain an upper bound of the H ∞ norm as a function of the H 2 norm, both in the continuous and discrete time cases. It is also shown that, in continuous time, the H 2 norm can be bounded by a function of the H ∞ norm and the bandwidth of the system.

A new proof of the discrete-time LQG optimal control theorems

Presents a unifying new proof for the three discrete-time linear quadratic Gaussian problems (deterministic, stochastic full information, and stochastic partial information) based on pathwise (deterministic) optimization. The essential difference between the control aspect of the three cases is that the controls should lie in different classes of admissible controls, and the authors address them as constrained optimization problems using appropriate Lagrange multiplier terms. >