Optimal multivariable controller design using an ITSE performance index

This article deals with the design of optimal controllers and, as an embedded problem, with the computation of achievable performance bounds in the control of non-minimum phase, unstable multi-input multi-output systems. The cost function comprises a time-weighted measure of the tracking error (ITSE) and an incremental quadratic penalisation of the control effort. The proposed methodology relies on the properties of a frequency domain generalisation of the ITSE index. The solution to the optimisation problem can be computed using orthonormal basis function expansions, and a closed-form expression is found for the optimal coefficients of the expansion. Numerical examples are presented to illustrate the results.     

PID controller design for fractional-order systems with time delays

Classical proper PID controllers are designed for linear time invariant plants whose transfer functions are rational functions of s^α, where 0<α<1, and s is the Laplace transform variable. Effect of input-output time delay on the range of allowable controller parameters is investigated. The allowable PID controller parameters are determined from a small gain type of argument used earlier for finite dimensional plants.     

Adaptive wavenet controller design for teleoperation systems with variable time delays using singular perturbation method

The main goal of controller design in teleoperation systems is to achieve stability and optimal operation in presence of factors such as time delays, system disturbances and modeling errors. This paper proposes a new method of controller design based on wavenet with singular perturbation method for the bilateral teleoperation of robots through the internet. The wavenet controller could overcome the variable time delay in teleoperation system. This new method introduces a reduced-order structure for control and stability of teleoperation systems. By using singular perturbation method, teleoperation system is decomposed into two fast and slow subsystems. This method is a step towards reduced-order modeling. In this method, we use a feedback linearization method in master subsystem and a wavenet controller for slave subsystem. In wavenet controller, we used a learning method so that the system was Lyapunov stable. As the stability of the model is highly dependent on the learning of the system, we use Lyapunov stability in this method. It has been tried to reduce the tracking error between the master and the slave subsystems. In this structure the position of master-slave are compared together and controlling signal is applied to the slave so that they can track each other in the least possible time. In all schemes the effectiveness of the system is shown through the simulations and they have been compared with each other.     

Decoupling controller design for linear multivariable plants

Decoupling controller design for linear time-invariant square multiple-input multiple-output (MIMO) plants under the unity-feedback configuration is discussed. For plants with no coincidences of unstable poles and zeros, a simplified necessary and sufficient condition for closed-loop stability is given. The simplified condition leads to a simple parameterization of the set of all achievable decoupled input/output (I/O) maps and an algorithm which allows the design of decoupling controllers to achieve preassigned closed-loop poles     

Optimal control of multivariable systems with pure time delays

A brief review of the development of the theory and computational approaches for the optimal control of multivariable systems having pure time delays is given. Following this is a detailed discussion of the optimal feedback control of both linear and non-linear systems having pure time delays. A neighboring optimal feedback control scheme is developed for non-linear problems, and a chemical reactor control problem is worked to illustrate the performance of the controller.     

A multivariable generalized self-tuning controller with decoupling design

This note describes a multivariable generalized self-tuning controller with decoupling which combines the classical pole-zero placement strategy with the optimal strategy for self-tuning control. The controller uses techniques of updating weighting polynomial matrices on line and eliminating the tracking errors without use of an integrator, and decouples the multivariable system.     

Non-fragile multivariable PID controller design via system augmentation

In this paper, the issue of designing non-fragile H_∞ multivariable proportional-integral-derivative (PID) controllers with derivative filters is investigated. In order to obtain the controller gains, the original system is associated with an extended system such that the PID controller design can be formulated as a static output-feedback control problem. By taking the system augmentation approach, the conditions with slack matrices for solving the non-fragile H_∞ multivariable PID controller gains are established. Based on the results, linear matrix inequality -based iterative algorithms are provided to compute the controller gains. Simulations are conducted to verify the effectiveness of the proposed approaches.     

Evolutionary algorithms based design of multivariable PID controller

In this paper, performance comparison of evolutionary algorithms (EAs) such as real coded genetic algorithm (RGA), modified particle swarm optimization (MPSO), covariance matrix adaptation evolution strategy (CMAES) and differential evolution (DE) on optimal design of multivariable PID controller design is considered. Decoupled multivariable PI and PID controller structure for Binary distillation column plant described by Wood and Berry, having 2 inputs and 2 outputs is taken. EAs simulations are carried with minimization of IAE as objective using two types of stopping criteria, namely, maximum number of functional evaluations (Fevalmax) and Fevalmax along with tolerance of PID parameters and IAE. To compare the performances of various EAs, statistical measures like best, mean, standard deviation of results and average computation time, over 20 independent trials are considered. Results obtained by various EAs are compared with previously reported results using BLT and GA with multi-crossover approach. Results clearly indicate the better performance of CMAES and MPSO designed PI/PID controller on multivariable system. Simulations also reveal that all the four algorithms considered are suitable for off-line tuning of PID controller. However, only CMAES and MPSO algorithms are suitable for on-line tuning of PID due to their better consistency and minimum computation time.     

A multivariable bilinear adaptive controller with decoupling design

This paper presents a new adaptive decoupling controller for multivariable bilinear systems with non-diagonal coefficient matrix of stochastic noises. The controller combines the feedforward control strategy with the generalized minimum variance approach and performs the decoupling of the control loop dynamically as well as in the steady state. It can control the bilinear systems whose linear part is not necessarily of minimum phase. The proof of global convergence for the algorithm is also provided. Simulation results demonstrate the effectiveness of the controller     

Self-tuning controller design for systems with arbitrary time delays Part 1. Theoretical development

Self-tuning control of multivariable systems with arbitrary time delays has been an active area of research in recent years. The use of an interactor matrix in deriving controllers for such systems has been suggested by various authors. Factorization of the input polynomial matrix has also been proposed for the controller design stage. It is shown in this paper that these two approaches are not, in general, equivalent. A simple and straightforward procedure for the extraction of the input and output time delay matrices is proposed for the input polynomial matrix factorization approach. Conditions for the time delay matrices to exist are also derived for a class of multivariable systems. These conditions provide a theoretical justification for the use of time delay matrices in self-tuning control of such systems.     

Constrained discrete-time PI controller design for output PDFs of stochastic systems with time delays

This article presents a new proportional-integral (PI) tracking control strategy for non-Gaussian stochastic systems based on a square root B-spline model for the output probability density functions (PDFs). Following the square root B-spline approximation to the measured output PDF, a non-linear discrete-time dynamical model can be established between the control input and the weights related to the PDFs. It is noted that the PDF tracking is transformed to a constrained dynamical tracking control problem for weight dynamics. For the non-linear discrete-time weight model including time-delay terms and exogenous disturbances, convex linear matrix inequality optimisation algorithms are used to design a generalised PI controller such that stabilisation, state constraint and tracking performance can be guaranteed simultaneously. Furthermore, in order to enhance the robustness, the peak-to-peak measure index is applied to optimise the tracking performance. Simulations are given to demonstrate the efficiency of the proposed approach.     

On the decoupling of multivariable control systems with time delays†

The decoupling problem is solved for an m-input m-output linear time-invariant system with multiple delays in the state and/or the control. The conditions under which such a system can be decoupled are found and the class of the feedback decoupling operators is given. These operators are realized by means of digital filters. Then, the more general linear model in which the input effects directly the output is considered which, upon the introduction of both feedback and feedforward, is shown to be decoupled, Two illustrative examples are included to support the theoretical results.     

Sliding mode observer–controller design for uncertain Markovian jump systems with time delays

This paper proposes a sliding mode observer–controller design method for uncertain Markovian jump systems with time delays and uncertain switching probabilities. Both the structures of a sliding mode observer and a sliding mode controller are given. By the mode‐dependent Lyapunov functional approach, a sufficient condition for the stochastic stability of the closed‐loop system is given, which can be converted into a convex optimization problem. The reachability of the sliding surfaces in both the estimation error space and the state estimate space can be ensured by the presented control scheme. Finally, the effectiveness of the proposed design method is demonstrated by a simulation example. Copyright © 2011 John Wiley & Sons, Ltd.     

Optimal uniform-damping ratio controller for sequential design of multivariable systems

An optimal uniform-damping ratio controller is developed for the sequential design of a multivariable control system so that the designed closed-loop poles of the respective multivariable system and reduced-order observer are exactly placed on the negative real axis and/or the boundaries of desired sectors with constant-damping ratios. The functions in the quadratic performance index to be minimized are chosen as a combination of the weighted outputs, reduced states and inputs. Also, the optimal uniform-damping ratio controller is a combination of optimal output-feedback and optimal reduced-order state-feedback controllers. A numerical example is given to demonstrate the design procedure.     

A multivariable on-line adaptive PID controller using auto-tuning neurons

In this paper, we present a new PID control technique based on auto-tuning neurons for multivariable systems. The main difference between an auto-tuning neuron and a general neuron is that there are adjustable parameters of the activation function used in an auto-tuning neuron. In this paper, a modified hyperbolic tangent function is used as the activation function of an auto-tuning neuron, which provides two adjustable parameters to flexibly determine the magnitude and the shape of function. We then use such auto-tuning neurons to find gains of the multivariable PID controller, which is tuned on-line according to certain adaptation laws. Finally, two illustrative examples will be used to compare the performance by using our proposed method and other methods.     

Tuning of a multivariable discrete time PI controller for unknown systems

Tuning of a multivariable discrete time PI controller, which uses interactions of the plant is discussed. Interactions are detected by observing the plant response to step inputs. The plant is assumed to be unknown, linear, time invariant and stable. The tuning method presented is an extension of the corresponding continuous time method (Davison 1976, Penttinen and Koivo 1980)     

A multivariable self-tuning controller

A multivariable self-tuning controller is derived extending the scalar version of Clarke and Gawthrop (1975). Because the control terms are penalized in the cost function, fluctuations and peaking in control signals are reduced compared with the multivariable minimum variance self-tuning controller (Borisson, 1979). Time-varying reference signals as well as certain nonminimum-phase systems can be handled without difficulty with the proposed controller. Several examples illustrate the power of the derived self-tuning controller.     

The optimal design of a discrete-time controller for a single-input,single-output system with time delays

The design of an optimal controller for a discrete-time single-input, single-output system is well documented in a number of textbooks (e.g. Bryson and Ho 1969). It has been shown by Hem (1969), using a procedure similar to that proposed by Lee (1964), and later by Bryson and Ho. that the optimal controller may be represented by its z-transfer function and therefore may be simply implemented as the ratio of two polynomials in z_?1, thus reducing appreciably the amount of on-line computation. Unfortunately, the method of Lee for the computation of the coefficients of the z-transfer function has been found to break down when the structure of the plant includes a finite time delay which creates a singularity in the plant transition matrix. In this paper it is shown that the existence of the singularity induces an orthogonality in the structure of the optimal controller equations which invalidates the application of Lee's method. An alternative, computationally robust, technique is presented for the computation of the coefficients of the z-transfer function of the optimal controller.     

Self-tuning controller design for systems with arbitrary time delays Part 2. Algorithms and simulation examples

Recent theoretical developments in the self-tuning control of multivariable systems with arbitrary time delays have been considered in Part 1 of this work (Tade et al. 1988) and a simple time delay extraction procedure has been proposed. In this paper, that procedure is used to derive an adaptive controller based on the generalized minimum variance criterion. The computation count for the proposed self-tuning control algorithm is compared with those for two other algorithms, one proposed by Prager and Wellstead (1981) and the other by Dugard et al. (1984). Assuming that all computation requirements for other stages of the implementation of these algorithms are equal, the results indicate that the proposed algorithm requires less computational effort for updating the control action at each sampling interval. Simulation examples are presented to demonstrate the performance of the proposed algorithm.