Functional relationships between the conventional steady-state error characteristics and the weighting matrices in the quadratic performance index†

The requirements for achieving a desired value of steady-state error for an optimal control system utilizing the quadratic performance index with a linear plant arc derived for step, ramp and parabolic inputs. These requirements result in a number of the elements of the Riccati matrix being specified in terms of known parameters. It is shown that for an all-pole plant the control system cannot follow a parabolic input. For the stop input the q ₁₁ element of the Q matrix is specified as one for an all-pole plant or in terms of the fixed plant parameters for a pole-zero plant. Depending on the number of requirements and the order of the system it may be necessary to assume values for the remaining elements of the Q matrix in order to solve the reduced Riccati equation.     

On modifying the prediction control of a second-order plant†

The switching behaviour of a second-order plant under the combined influence of system non-linearities such as Coulomb friction, stiction, backlash, dead zone, hysteresis and switching delay is analytically examined in this paper.

In the first instance, the mathematical model which characterizes the system dynamics is formulated on a more realistic basis by considering the effect of these non-linearities as integrated parameters rather than isolated phenomena. This is then followed by the derivation of the modified switching equation which, upon careful mathematical manipulation, may be expressed in terms of a power series as a function of the system velocity. Such an approach not only leads to a better understanding of the general switching behaviour during the change over of forward to reversed drive, but also provides a more practical solution for the bang-bang control problem     

A discretization of the cost functional in linear-quadratic optimal control†

A discretized cost functional in linear-quadratic control is presented. Specifically, a hybrid linear-quadratic optimal control problem is analysed from the feedback, open-loop and existence-of-solutions points of view, methods being employed from the well established theory for the solution of linear tridiagonal systems of equations.     

On the discrete time matrix Riccati equation of optimal control†

This paper is concerned with the discrete time matrix Riccati equation. The properties established are those of minimality, convergence, uniqueness and stability. Further the convergence of the policy space approximation technique is proved. These results are analogous to those known for the continuous-time Riccati equation, but the techniques used are simpler.     

A simple iterative method for sub-optimal control of linear time delay systems with quadratic cost†

The usual approach for obtaining the optimal control For a linear time-delay system with a quadratic cost consists of first deriving a set of necessary conditions for optimality and then using conventional iterative numerical methods to find a control satisfying those conditions. The burden of computation in this approach is enormous. The iterative scheme proposed in this paper does not proceed along these lines. Instead, the delay term is treated liko an extra perturbing input. A linear non-delay system is optimized at each stage, and the resulting sequence of control functions converges rapidly to the sub-optimal control for the original problem, as illustrated by two numerical examples.     

On the design of optimal and sub-optimal control systems†

Though dynamic programming is an excellent and versatile algorithm, there are many cases in which we cannot apply this algorithm favourably to control system design because of the limited capacity of a digital computer and other difficulties.

Considering these facts, the successively optimizing method and the combined algorithm of dynamic programming and successively optimizing method are presented in this paper.

One of the objects of this paper is to present a practical method for obtaining the optimal solution of control systems with an integral form performance index by the use of dynamic programming.

The other is to present sub-optiinal solutions by the successively optimizing method and the combined algorithm.

These methods are very easy when compared with dynamic programming. Furthermore, these sub-optimal solutions can be made to converge to the optimum by using the recurrent equations repeatedly.     

Weighted residual methods and the suboptimal control of distributed parameter systems†

A suboptimal control algorithm for distributed parameter systems is developed in a framework which synthesizes weighted residual methods and mathematical programming. The heat exchanger example of Koppel et al. (1968) is employed for introducing the algorithm. First, the Galerkin procedure with polynomial modes is applied to obtain a lumped ODE model for the distributed parameter system. Then the state and control variables of the lumped control problem are approximated by cubic splines on a uniform mesh. Through collocation at the knots, the ODE model is reduced to a sot of linear algebraic equations and the suboptimal control is determined from the solution of a quadratic programming problem with sparse matrices.

Numerical results for the heat exchanger example are presented and compared with those obtained by the authors (Neuman and Sen 1972) using the Ritz-Trefftz algorithm (Bosarge and Johnson 1970) for the lumped control problem. For this example, the two algorithms yield essentially identical results with comparable computational requirements. Application of the Ritz-Trefftz algorithm, however, is limited to lumped, linear-quadratic control problems without constraints on the state or control. The approach advocated in this paper, therefore, offers a viable approach to control problems in distributed parameter systems.     

D.C. gain of the non-linear element and the describing function†

In the present paper it has been shown that the maximum value of the describing function is related to the d.c. gain of the non-linearity, and the describing function analysis, where applicable, gives more information regarding system stability than is available from Aizerman's conjecture.     

Continuous-time singular linear–quadratic control: Necessary and sufficient conditions for the existence of regular solutions

The purpose of this paper is to provide a full understanding of the role that the constrained generalized continuous algebraic Riccati equation plays in singular linear–quadratic (LQ) optimal control. Indeed, in spite of the vast literature on LQ problems, only recently a sufficient condition for the existence of a non-impulsive optimal control has for the first time connected this equation with the singular LQ optimal control problem. In this paper, we establish four equivalent conditions providing a complete picture that connects the singular LQ problem with the constrained generalized continuous algebraic Riccati equation and with the geometric properties of the underlying system.     

On the relation between the continued fraction expansion and moments matching methods of model reduction †

A widely used form of transfer function model is $ Two methods of approximating such transfer function models by lower order models, the continued fraction expansion of Chen and Shieh and the moments matching method of Gibilaro and Lees, are compared. It is demonstrated that these two methods can give identical results. An efficient computational procedure for the moments matching method is described and the results of computation for the reduction of a seventh-order model are given.     

2–2 composites based on [011]-poled relaxor-ferroelectric single crystals: analysis of the piezoelectric anisotropy and squared figures of merit for energy harvesting applications

In this paper we explore the effect of the orientation of the main crystallographic axes in relaxor-ferroelectric single crystals (SCs) on the piezoelectric anisotropy and squared figures of merit of 2–2 parallel-connected SC/auxetic polymer composites. The single-crystal component for the composite is chosen from the perovskite-type solid solutions with compositions near the morphotropic phase boundary and poled along the perovskite unit-cell [011] direction (mm ² symmetry of domain-engineered SCs). The orientation of the main crystallographic axes in the single-crystal component is observed to strongly influence the piezoelectric coefficients $d_{3j}^{*}$ , squared figures of merit $d_{3j}^{*}$ $g_{3j}^{*}$ , electromechanical coupling factors $k_{3j}^{*}$ , and hydrostatic analogs of these parameters of the 2–2 composite. Inequalities $| {d_{33}^{*} /d_{3f}^{*} } | > 5$ and $| {k_{33}^{*} /k_{3f}^{*} } | > 5$ (f = 1 and 2) are achieved at specific orientations of the main crystallographic axes due to the significant anisotropy of the elastic and piezoelectric properties of the single-crystal component. The use of an auxetic polyethylene (a polymer component with a negative Poisson’s ratio) leads to a significant increase in the hydrostatic parameters. Particular advantages of such composites over conventional ceramic/polymer composites are taken into account for transducer, hydroacoustic, energy harvesting, and other applications.     

Sampled-data theory applied to the modelling and control analysis of compression ignition engines—Part I†

Proposals for the use of electronic speed governing of diesel engines used for driving alternators has led to a re-examination of the dynamic behaviour of the engine for control purposes. If maximum advantage to the plant performance is to be achieved from using more complex and expensive governing devices, then better models of individual components in the power system for simulation and control studies may be necessary. This paper describes an investigation performed to model the engine in terms of a discrete system. The frequency domain properties of the model (for various numbers of cylinders) are examined and compared with the model usually assumed, as well as with the frequency response of an actual engine. A companion paper will extend this work using 2-transform techniques.     

Feedback control of uncertain systems: robustness with respect to neglected actuator and sensor dynamics†

A purely algebraic polynomial approach is used to optimize the discrete-time feedback controller for a multivariable plant subject to stochastic reference and disturbance and coloured output measurement noise. A quadratic cost criterion with dynamic weightings is used that includes sensitivity and complementary sensitivity as well as tracking error and control input. The method involves the solution of only one diophantine equation.     

Applicability of location compatibility to the arrangement of display and control in human – vehicle systems: Comparison between young and older adults

The effects of age on applicability of the location compatibility principle to the design of display and control systems were discussed. A dual-task experiment was conducted, in which the primary task was first-order tracking. The secondary tasks included control of an air conditioner, the operation of a radio and the operation of a CD/MD, by means of either a steering wheel-mounted switch or a console-mounted switch. The display was arranged either in front of or on the left side of a participant. Performance of the young group did not completely follow predictions of the compatibility principle. In particular, the principle did not apply to the left-side display condition. The steering wheel-mounted switch was more effective than the left-side switch even for the left-side display. The compatibility principle was applicable to both front and left-side displays for the older adults. For the front display, the steering wheel-mounted switch was more effective. Such differences should be taken into account when designing display and control systems in man – vehicle systems.     

Systematic H∞ weighting function selection and its application to the real-time control of a vertical take-off aircraft

In this paper, the selection of H_∞ weighting functions for general practical applications is investigated. It is shown that an H_∞ weighting function for a single-input–single-output (SISO) plant can be obtained by considering a series of connections of elementary low-order plants. For a constrained control effort, an explicit weighting function expression is derived for the first time. A novel method for the selection of weighting functions in H_∞ mixed sensitivity design is proposed to control the percentage overshoot directly. An example illustrating how the transient response of a standard H_∞ design is improved by using the proposed weighting functions is provided. Finally, real-time experimental results are presented for the roll-angle control of a laboratory-scale physical model of a vertical take-off aircraft.     

The open question of the relation between square matrix’s eigenvalues and its similarity matrix’s singular values in linear discrete system

In linear discrete system, we often need to utilize similarity transformation to extend the solution ranges of its corresponding discrete matrix equations. Consequently, how to choose the similarity transformation matrix remains an open question in recent years. In this paper, by applying the theory of matrix’s Jordan canonical form and the related properties of nonnegative matrices, we construct the similarity transformation matrix of some special similarity transformation, then present a necessary and sufficient condition and a corresponding algorithm, thus solve the open question totally.     

Predictive control of an activated sludge process: An application to the Viikinmäki wastewater treatment plant

In this work, we discuss the application of multivariable predictive control for the activated sludge process in a full-scale municipal wastewater treatment plant. Emphasis is given to the selection of a control configuration that contributes to minimising the economic costs while improving the removal efficiency of the nitrogen compounds. For this task, a simple dynamic matrix control algorithm is favoured for controlling the nitrogen concentrations at the end of the biological process. The behaviour of the activated sludge process is reproduced in a commercial simulator that acts as a real-time testing platform and that is also used for identifying the multivariable input–output models for the predictive controller. For demonstrating the effectiveness of the proposed approach, different control configurations are considered and compared against the aeration control strategies currently used at the plant. Based on the simulation results, this work shows the potentiality of the dynamic matrix control which is able to decrease the energy consumption costs and, at the same time, reduce the ammonia peaks and nitrate concentration in the effluent.     

On the optimal control of systems having pure time delays and singular arcsI,Some necessary conditions for optimality†

The problem of singular arcs in the optimal control of systems having time delays is discussed. An expanded Legendre-Clebsch necessary condition is derived, and junction conditions for the boundary between non-singular and singular arcs is presented for a general class of problems. These and other ideas are illustrated by a number of examples.