Reduced‐order LQG/LTR procedure for the plant output side

It is commonly believed that reduced‐order observers, including reduced‐order Kalman filters, cannot be used in the loop transfer recovery (LTR) design of the plant output side. In contrast to common understanding, we show that, at least for nonminimum‐phase plants, the reduced‐order Kalman filter can be used in the linear‐quadratic‐Gaussian (LQG)/LTR design of the plant output side with clear meaning in systems theory. The key concept is to regard a reduced‐order Kalman filter as a high‐gain full‐order Kalman filter. For the reduced‐order LQG controller, we examine the asymptotic property achieved by applying the recovery procedure used in the full‐order LQG/LTR design. Using the equivalent full‐order Kalman filter, we find that the sensitivity property of the reduced‐order LQG controller is asymptotically equivalent to that of a high‐gain partial output injection system. Motivated by this result, we propose the reduced‐order LQG/LTR procedure taking the high‐gain partial output injection system as a target. Some target properties are discussed to clarify the difference from the full‐order design. A multivariable design example is presented to show that the procedure provides a systematic design of a reduced‐order controller with optimality consideration.     

Reduced‐order observer‐based fault estimation and fault‐tolerant control for a class of discrete Lipschitz nonlinear systems

This paper addresses an issue on reduced‐order observer‐based robust fault estimation and fault‐tolerant control for a class of uncertain nonlinear discrete‐time systems. By introducing a nonsingular coordinate transformation, a new nonlinear reduced‐order fault estimation observer (RFEO) is proposed with a wide application range in order to achieve an accurate estimation of both states and faults. Next, an improved algorithm is given to obtain the optimal estimation by using a novel iterative linear matrix inequality technique. Furthermore, an RFEO‐based output feedback fault‐tolerant controller, which is independent of the RFEO, can maintain the stability and performance of the faulty system. Simulation results of an aircraft application show the effectiveness of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Minimizing control energy in a class of bounded‐control linear‐quadratic regulator problems

Minimal‐control‐energy strategies are substantiated and illustrated for linear‐quadratic problems with penalized endpoints and no state‐trajectory cost, when bounds in control values are imposed. The optimal solution for a given process with restricted controls, starting at a known initial state, is shown to coincide with the saturated solution to the unrestricted problem that has the same coefficients but starts at a generally different initial state. This result reduces the searching span for the solution: from the infinite‐dimensional set of admissible control trajectories to the finite‐dimensional Euclidean space of initial conditions. An efficient real‐time scheme is proposed here to approximate (eventually to find) the optimal control strategy, based on the detection of the appropriate initial state while avoiding as much as possible the generation and evaluation of state and control trajectories. Numerical (including model predictive control) simulations are provided, compared, and checked against the analytical solution to ‘the cheapest stop of a train’ problem in its pure‐upper‐bounded brake, flexible‐endpoint setting. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

The optimal feedback control of the linear‐quadratic control problem with a control inequality constraint

In this paper, we consider the linear‐quadratic control problem with an inequality constraint on the control variable. We derive the feedback form of the optimal control by the agency of the unconstrained linear‐quadratic control systems. Copyright © 2001 John Wiley & Sons, Ltd.     

Near‐optimal control for multiparameter singularly perturbed stochastic systems

In this paper, the linear quadratic optimal stochastic control problem is investigated for multiparameter singularly perturbed stochastic systems in which N lower‐level fast subsystems are interconnected by a higher‐level slow subsystem. After establishing the asymptotic structure of the solution for the multiparameter stochastic algebraic Riccati equation (MSARE), a near‐optimal controller that is independent of small unknown parameters is obtained by neglecting these parameters. The stability of a closed‐loop stochastic system is investigated. Furthermore, it is shown that the resulting controller achieves an O(∥ν∥²) approximation to the optimal cost of the original optimal control problem. Finally, in order to demonstrate the efficiency of the proposed algorithm, a numerical example—a practical multi‐area power system—is solved. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimal control for discrete‐time singular stochastic systems with input delay

The finite‐horizon linear quadratic regulation problem is considered in this paper for the discrete‐time singular systems with multiplicative noises and time delay in the input. Firstly, the extremum principle is discussed, and a stationary condition is derived for the singular stochastic system. Then, based on the relationships established between the state and the costate variables, the stationary condition is also shown to be a sufficient criterion assuring the existence of the solution for the stochastic control problem. The optimal controller is designed as a linear function of the current state and the past inputs information, which can be recursively calculated by effective algorithms. With the designed optimal controllers, the explicit expression is also derived for the minimal value of the performance index. One numerical example is provided in the end of the paper to illustrate the effectiveness of the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.     

An efficient scheme for minimax solutions of multiple linear‐quadratic control

Optimal control is one of the most important methodologies for studies of dynamic systems in many areas of sciences, engineering and economics. Minimax optimal control is a special topic in the general framework of multiple optimal control problems. Minimax optimal control can be considered as a dynamic game with multiple players under the same system. In this paper, we develop a fast search for a minimax solution of multiple linear‐quadratic control problems. The algorithm improves the existing solution scheme by adjusting the multiple weighting coefficients in each iteration and also including updates for step‐size control. Copyright © 2005 John Wiley & Sons, Ltd.     

On the synthesis of time‐varying LQG weights and noises along optimal control and state trajectories

A general approach to control non‐linear uncertain systems is to apply a pre‐computed nominal optimal control, and use a pre‐computed LQG compensator to generate control corrections from the on‐line measured data. If the non‐linear model, on which the optimal control and LQG compensator design is based, is of sufficient quality, and when the LQG compensator is designed appropriately, the closed‐loop control system is approximately optimal. This paper contributes to the selection and computation of the time‐varying LQG weighting and noise matrices, which determine the LQG compensator design. It is argued that the noise matrices may be taken time‐invariant and diagonal. Three very important considerations concerning the selection of the time‐varying LQG weighting matrices are turned into a concrete computational scheme. Thereby, the selection of the time‐varying LQG weighting matrices is reduced to selecting three scalar design parameters, each one weighting one consideration. Although the three considerations seem straightforward they may oppose one another. Furthermore, they usually result in time‐varying weighting matrices that are indefinite, rather than positive (semi) definite as required for the LQG design. The computational scheme presented in this paper addresses and resolves both problems. By two numerical examples the benefits of our optimal closed‐loop control system design are demonstrated and evaluated using Monte Carlo simulation. Copyright © 2005 John Wiley & Sons, Ltd.     

High‐order observers‐based LQ control scheme for wind speed and uncertainties estimation in WECSs

Tip speed ratio control is a popular method in wind energy conversion systems in order to capture the maximum power. This method, however, requires wind speed information, which is difficult in practice to accurately measure it. Therefore, estimation methods are usually applied, where a high‐precision estimate leads to a high‐efficient system. Based on the fact that the wind speed varies in a random way, this paper proposes a generalized high‐order observer to estimate the aerodynamic torque and the wind speed accordingly. This observer algorithm releases the assumption that the wind speed should be slowly varying, which is required in previous observer designs. Moreover, two other generalized high‐order observers are also applied to estimate the uncertainties, which depend on state variables and cannot be considered as slow‐varying disturbances. Using the outputs of these observers, a robust high‐performance optimal control system is developed for the rotor speed to keep the optimal tip speed ratio. The stability analysis of the designed control system is fully presented. The effectiveness of the proposed technique is validated via simulation studies.     

Stochastic linear quadratic optimal control problem for systems driven by fractional Brownian motions

This paper is concerned with stochastic linear control systems driven by fractional Brownian motions (fBms) with Hurst parameter H∈(1/2,1) and the cost functional is quadratic with respect to the state and control variables. Here, the integrals with respect to fBms are the type of Stratonovich integrals. A stochastic maximum principle as a necessary condition of the optimal control is derived. The adjoint backward stochastic differential equation (BSDE) is driven by the fBms and its underlying standard Brownian motions. The existence and uniqueness of the solution of adjoint BSDE is proved. The explicit form of the unique optimal control is obtained.     

Partially observed time‐inconsistent stochastic linear‐quadratic control with random jumps

This paper studies a partially observed time‐inconsistent stochastic linear‐quadratic control system, in which the state follows a stochastic differential equation driven by a Brownian motion and an independent Poisson random measure. The cost functional contains a state‐dependent term and a quadratic term of the conditional expected state process, which will cause the time inconsistency in dynamic systems. By virtue of a classical spike variation approach, we define an equilibrium and derive a sufficient condition for the equilibrium in the fully observed system with stochastic coefficients. Then, we obtain the equilibrium with an explicit feedback form in deterministic coefficients case and discuss the existence and uniqueness of the solution of corresponding Riccati equations. Furthermore, we get filtering equations of the partially observed system and get the explicit equilibrium in some special case.     

Optimal multi‐interval control of a cantilever beam by a recursive control algorithm

The optimal‐distributed control of a transversely vibrating cantilever beam is studied with the objective of minimizing the deflection and velocity in a given period of time with the minimum possible expenditure of force. The beam undergoes transient vibrations and is subject to given displacement and velocity initial conditions. The control is exercised by means of a transversely distributed force referred to as the control force. In the present study, a multi‐interval optimal control method is developed with the application of a maximum principle. The method consists of dividing the control duration into several intervals and using the maximum principle to obtain the optimality conditions at each interval. The explicit solutions for a cantilever beam are obtained by a recursive algorithm that takes the final conditions of the last interval as the initial conditions of the next interval. The formulation and the method of solution are suitable and convenient for digital computation. Numerical results are given, which compare the deflections, velocities and the control force under the optimal multi‐interval control with those under the optimal single‐interval control. Copyright © 2008 John Wiley & Sons, Ltd.     

Implementation lessons and pitfalls for real‐time optimal control with stochastic systems

Modern computational power and efficient direct collocation techniques are decreasing the solution time required for the optimal control problem, making real‐time optimal control (RTOC) feasible for modern systems. Current trends in the literature indicate that many authors are applying RTOC with a recursive open‐loop structure, relying on a high recursion rate for implicit state feedback to counter disturbances and other unmodeled effects without explicit closed‐loop control. The limitations of using rapid, instantaneous optimal solutions are demonstrated analytically and through application to a surface‐to‐air missile avoidance control system. Two methods are proposed for control structure implementation when using RTOC to take advantage of error integration through either classical feedback or disturbance estimation. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.     

Optimal control of non‐linear chemical reactors via an initial‐value Hamiltonian problem

The problem of designing strategies for optimal feedback control of non‐linear processes, specially for regulation and set‐point changing, is attacked in this paper. A novel procedure based on the Hamiltonian equations associated to a bilinear approximation of the dynamics and a quadratic cost is presented. The usual boundary‐value situation for the coupled state–costate system is transformed into an initial‐value problem through the solution of a generalized algebraic Riccati equation. This allows to integrate the Hamiltonian equations on‐line, and to construct the feedback law by using the costate solution trajectory. Results are shown applied to a classical non‐linear chemical reactor model, and compared against suboptimal bilinear‐quadratic strategies based on power series expansions. Since state variables calculated from Hamiltonian equations may differ from the values of physical states, the proposed control strategy is suboptimal with respect to the original plant. Copyright © 2005 John Wiley & Sons, Ltd.     

Analytical design of stable delta‐domain generalized predictive control

This paper addresses certain fundamental issues related to the discrete‐time design problem of the delta‐domain generalized predictive control (δ‐GPC) for both minimum phase and non‐minimum phase linear SISO plants including nominal stability and nominal performance of the closed‐loop system. The approach being presented is completely analytical, and the nominal performance of the control system is directly achieved by a prototype design of the closed‐loop system characteristics resulting in definite time‐domain specifications. Two design methods are offered in which a model‐based prediction paradigm is applied to achieve the future output and the future filtered output trajectory of the plant. Prediction of the first type is based on suitable emulations of the output δ‐derivatives and is used in the GPC controller design for minimum‐phase models of the plant. Prediction of the second type utilizes emulation of derivatives of the output filtered by the numerator polynomial of the transfer function of the controlled part of the plant. It can be employed both for minimum phase and non‐minimum phase plants. A numerical example is given that illustrates the δ‐GPC method for controller design. Copyright © 2002 John Wiley & Sons, Ltd.