Quadratic optimal controller to stabilise symmetrical systems

In this paper, we design a controller for stabilising a control system. The technique used for designing the controller includes a linear regulator and an asymptotical observer which form the controller. The linear regulator designed is a feedback of estimated states and also it must minimise a quadratic performance index. The gain matrix of optimal feedback is obtained by solving the Riccati equation, whilst the gain observer matrix is computed by making use of symmetrical systems properties. The properties of symmetrical systems allow us to find the optimal gain matrix of the observer without solving the dual Riccati equation, we only need to compute the matrices of controllability and observability. Having calculated the gain matrices of regulator and of observer, we proceeded to compute the transfer function of the observer-based controller.     

Guaranteed cost nonlinear tracking control of a boiler-turbine unit: an LMI approach

This article addresses the problem of designing a guaranteed cost nonlinear state feedback tracking control for a boiler-turbine unit. First, the nonlinear boiler-turbine is re-expressed as a linear system with norm bounded uncertainties via a nonlinear transformation function. Then, based on this linear model a sufficient condition for the existence of a guaranteed cost nonlinear state feedback tracking control is derived in terms of linear matrix inequalities. The advantage of the proposed tracking control design is that only a simple nonlinear controller is constructed and it does not involve feedback linearisation technique and complicated adaptive or fuzzy schemes. An industrial boiler-turbine system is used to illustrate the effectiveness of the proposed design as compared with a linearised approach.     

Design of fuzzy controllers for uncertain nonlinear systems using stability and robustness analyses

This paper analyzes the stability and robustness of uncertain nonlinear systems and shows that the analysis results provide an efficient technique for the design of fuzzy controllers. Based on a fuzzy plant model describing an uncertain nonlinear plant, this design involves the derivation of a stability criterion and a robust area in the uncertain parameter space in terms of some measures of the closed-loop control system matrices. An application example on balancing an inverted pendulum is given to illustrate the simple design methodology, the stability and the robustness of the feedback system incorporated with the proposed fuzzy controller.     

An approximated scalar sign function approach to optimal anti-windup digital controller design for continuous-time nonlinear systems with input constraints

This article presents an approximated scalar sign function-based digital design methodology to develop an optimal anti-windup digital controller for analogue nonlinear systems with input constraints. The approximated scalar sign function, a mathematically smooth nonlinear function, is utilised to represent the constrained input functions, which are often expressed by mathematically non-smooth nonlinear functions. Then, an optimal linearisation technique is applied to the resulting nonlinear system (with smooth nonlinear input functions) for finding an optimal linear model, which has the exact dynamics of the original nonlinear system at the operating point of interest. This optimal linear model is used to design an optimal anti-windup LQR, and an iterative procedure is developed to systematically adjust the weighting matrices in the performance index as the actuator saturation occurs. Hence, the designed optimal anti-windup controller would lie within the desired saturation range. In addition, the designed optimal analogue controller is digitally implemented using the prediction-based digital redesign technique for the effective digital control of stable and unstable multivariable nonlinear systems with input constraints.     

On the nature of block-pulse operational matrices: some further results

The paper presents some further results concerning the methods of the block-pulse functions. Based on an earlier paper, it is shown, that the technique of the block-pulse operational matrices can be simplified by considering its discrete nature and connections with the well-known Z transform. This new approach results in simple recursive algebraic equations and can be used successfully for linear and non-linear, time-invariant and time-varying systems. First, the methods of the block-pulse convolution and stretch matrices are replaced with the method of the so-called ‘block-pulse transform’. Next, the same technique gives the recursive algebraic equations for the non-linear differential equations. The form of the recursive equations depends on the choice of the particular type of block-pulse integrating matrices (conventional, generalized and so on). Then, the block-pulse transform is applied for solving varying systems with and without delays. Finally, some new results concerning the block-pulse technique of the inverse Laplace transform for irrational and transcendental transfer functions are given. A new theorem of the equivalence of the two methods is derived and proved. Many illustrative examples show the superiority of the new approach, which compared with the methods of block-pulse or Walsh matrices seems to be very attractive, simple in the construction and easily used in a computer program.     

Multi-objective controller design for discrete stochastic optimal systems with non-linear time-varying unmodelled dynamics and noise spectral uncertainties

A simple technique is introduced to synthesize an optimal controller, not only to minimize the least favourable cost functional J, but also to achieve the following purposes in discrete-time linear-quadratic-gaussian (LQG) optimal systems: (I) input-output decoupling; (2) stability robustness in the presence of non-linear time-varying (NLTV) unmodelled dynamics; (3) complete and arbitrary stable pole-placement; and (4) some zero-assignment, The Wiener-Hopí technique is employed and two weighting matrices Q(z) and R(z) of the quadratic cost functional are specified (by the inverse optimal control method), so that the controller is optimal with respect to the chosen weighting matrices and the design goals are achieved.     

An automated method for the preparation of orthogonal arrays for use in Taguchi designed experiments

In the last several years, the use of designed experiments in manufacturing and engineering design environments has become increasingly popular through the introduction of the ideas of Dr. G. Taguchi. The Taguchi Method, a systematic technique for experimental design and analysis, employs team oriented solutions to analyze design and production problems and their causes. The Taguchi Method as provided a simplified approach to the design of statistically significant experiments. This has greatly increased the number of experimental design practitioners.

This paper presents a computer program that addresses the need for an automated system able to design the experimental matrices for the orthogonal arrays that are required by Taguchi's Method. The program was written tode sign simple arrays as well as complex multi-level arrays with two-way interactions.     

Input decoupling with PD and preview control law for non-strictly proper systems

In this article, a solution is presented for the input decoupling problem with a control input consisting of a proportional-plus-derivative (PD) function of the signal to be rejected, with the requirement of internal stability of the closed-loop, in the general case where all the feedthrough matrices are possibly non-zero. This problem coincides with the so-called decoupling with preview in the discrete-time case. However, in this article the continuous and discrete cases are treated in a unified setting. A simple synthesis technique is also presented for the computation of the gain matrices appearing in the decoupling control law. This technique does not require the extra assumption of left invertibility, or changes of coordinates. In the last part of this article, the implementation of the decoupling control is discussed both via mixed feedback/feedforward and pure feedforward control architectures.     

Optimal linear filter design for systems with correlation in the measurement matrices and noises: recursive algorithm and applications

This paper addresses the optimal least-squares linear estimation problem for a class of discrete-time stochastic systems with random parameter matrices and correlated additive noises. The system presents the following main features: (1) one-step correlated and cross-correlated random parameter matrices in the observation equation are assumed; (2) the process and measurement noises are one-step autocorrelated and two-step cross-correlated. Using an innovation approach and these correlation assumptions, a recursive algorithm with a simple computational procedure is derived for the optimal linear filter. As a significant application of the proposed results, the optimal recursive filtering problem in multi-sensor systems with missing measurements and random delays can be addressed. Numerical simulation examples are used to demonstrate the feasibility of the proposed filtering algorithm, which is also compared with other filters that have been proposed.     

Design of state feedback for large-scale multivariable systems

This note presents a general method which reduces the computational requirements in the state feedback design of large-scale multivariable systems. The given system is first transformed into a general block canonical form by using simple equivalent transformation. The state feedback problem is then reformulated in terms of a Sylvester equation. Finally, the transformed system matrices along with certain assumed block forms for unknown matrices enable the Sylvester equation to be decomposed and solved effectively.     

Toeplitz matrices for LTI systems,an illustration of their application to Wiener filters and estimators

The Wiener–Kolmogorov theory of filtering has been with us since the first half of the twentieth century. A later matrix-based approach which was more general was derived with the steady-state Kalman filter. This approach uses a novel method of representing causal and uncausal systems in the form of convolution matrices and leads to a Wiener solution which is much easier to calculate than either the Kalman or Wiener approaches. For coloured additive noise, it avoids the use of Diophantine equations. The key idea missing in previous work is the close link between polynomials and Toeplitz matrices which are lower triangular in form. There is already a reasonably sized literature in the mathematics field on such matrices and so the area is ripe for exploration. Although the method does not offer a different or better solution, it shows a completely new way of defining linear time-invariant (LTI) systems which is neither transfer-function nor state-space-based. This is achieved by exploiting the connection between polynomials and Toeplitz matrices. The application here is the Wiener filter but there could well be many more as this is a generic approach.     

线性时滞系统滞后反馈鲁棒镇定

本文研究时不变线性时滞系统的鲁棒镇定问题。通过建立时滞系统的一个渐近稳定性定理，对摄动矩阵满足匹配条件和不满足匹配条件的情况分别给出了完全鲁棒镇定控制器的设计方法与鲁棒镇定控制器的存在性充分条件和设计方法；文中尤其提出了非滞后线性系统的一种简单的滞后反馈镇定方案。文末用例子示例了本文的设计方法。     

Global inverse optimal stabilization of stochastic nonholonomic systems

Optimality has not been addressed in existing works on control of (stochastic) nonholonomic systems. This paper presents a design of optimal controllers with respect to a meaningful cost function to globally asymptotically stabilize (in probability) nonholonomic systems affine in stochastic disturbances. The design is based on the Lyapunov direct method, the backstepping technique, and the inverse optimal control design. A class of Lyapunov functions, which are not required to be as nonlinearly strong as quadratic or quartic, is proposed for the control design. Thus, these Lyapunov functions can be applied to design of controllers for underactuated (stochastic) mechanical systems, which are usually required Lyapunov functions of a nonlinearly weak form. The proposed control design is illustrated on a kinematic cart, of which wheel velocities are perturbed by stochastic noise.     

Compensator design for linear multivariable systems

A configuration and an orderly procedure for the design of linear multivariable control systems are presented. This procedure employs transfer function matrices to specify the system response to set point changes and disturbances. Neither a state-variable formulation nor solution of the matrix Riccati equation are required. The design procedure is illustrated by a simple example involving an unstable system.     

Tensor Product‐Based Model Transformation and Optimal Controller Design for High Order Nonlinear Singularly Perturbed Systems

This paper investigates the optimization of a general class of nonlinear singularly perturbed systems. Tensor product‐based modeling, algebraic properties, and singular perturbation theory played a significant role in the formulation of the derived reduced model. The obtained reduced output is a nonlinear state and input dependent vector. In this case, solving the optimal control problem when minimizing a quadratic performance index becomes more difficult because the weighting matrices vary as functions of states. So the reformulation of the initial problem is needed and the resolution is discussed from an analytical point of view. A two‐step controller design procedure is suggested and an algorithm is proposed for the calculus of the gain matrices of the nonlinear control law. The Simulation results are presented to demonstrate the effectiveness of the approach and the power of the implemented algorithm.     

Decomposition and stability of linear systems with multiplicative noise

This paper concerns a representation of solutions and the stability of linear systems with multiplicative white noise, which is described by a vector Ito stochastic differential equation. The solution can be represented as a finite product of exponential matrices if Lie algebra generated by system matrices is solvable. If Lie algebra is not solvable, it is shown by the decomposition principle of Lie algebra that the problem of solving an equation can be reduced to the problem of solving a set of equations, whose corresponding Lie algebra is simple. Noting the structure of the sample solution, we present a technique of obtaining asymptotic stability conditions of sample solutions w.p.1, in the pth-order moment and in the pth-mean moment. The necessary and/or sufficient conditions of stability in some stochastic sense are obtained under certain conditions.     

Design and analysis of discrete-time robust Kalman filters

In this paper, the problem of finite and infinite horizon robust Kalman filtering for uncertain discrete-time systems is studied. The system under consideration is subject to time-varying norm-bounded parameter uncertainty in both the state and output matrices. The problem addressed is the design of linear filters having an error variance with an optimized guaranteed upper bound for any allowed uncertainty. A novel technique is developed for robust filter design. This technique gives necessary and sufficient conditions to the design of robust quadratic filters over finite and infinite horizon in terms of a pair of parameterized Riccati equations. Feasibility and convergence properties of the robust quadratic filters are also analyzed.     

Analysis of beams on BI-moduli elastic foundation

The analysis of beams on elastic foundation using the Winkler model is very common in engineering. When an elastic foundation is incapable of exerting tensile reactions, or when it has different stiffnesses in compression and tension, the simple Winkler model leads to a set of nonlinear equations which are complicated even for simple cases. Using exact stiffness matrices for beams on elastic foundation the problem is solved in an iterative technique, in which secondary nodes are inserted at transition points. Comparisons are made with simple case results in literature and the efficiency of the proposed procedure is demonstrated for a complex loading system.     

The application of parameter optimisation techniques to linear optimal control system design

The linear quadratic regulator (LQR) design technique produces controllers with well-known properties including good stability margins. A fundamental question with the LQR technique is the appropriate selection of the state and control weighting matrices to achieve given design specifications. A solution to this problem is presented whereby a parameter optimisation approach is employed to automate the LQR design procedure. Standard numerical algorithms are used to calculate the weighting matrices such that the specifications are satisfied. Decoupled command tracking specifications in the time domain are considered at present, and a multivariable proportional plus integral controller structure is employed. To the knowledge of the authors the design methodology is new and has the advantages of both optimal control and accommodation of additional design specifications. The application of the method is demonstrated in the design of a decoupled longitudinal control system for a vertical take-off-and-land (VTOL) aircraft.