Multivariable system synthesis with step disturbance rejection

Relying on the theory behind the Smith-MacMillan form of a rational matrix at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s = \infty</tex> , a necessary and sufficient condition is derived from the solvability of matrix Diophantine equations of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(s)X(s)+B(s)Y(s) = M(s)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(s), B(s)</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M(s)</tex> are given proper rational matrices and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X(s)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y(s)</tex> are unknown proper rational matrices. It is shown that the above result can be used in order to resolve in a new illuminating way the exact model matching problem (EMMP). If a solution to EMMP exists, then the family of all solutions is parametrized.

COMPUTER-AIDED DESIGN OF MULTIVARIABLE STOCHASTIC CONTROL SYSTEMS WITH A-PRIORI PRESCRIBED DYNAMICAL PROPERTIES

Structure at infinity and defect of transfer matrices with time-varying coefficients, with application to exact model-matching

Finite Time Model Matching for Multivariable Linear Systems

The present work proposes novel methods of Quenching self-sustained oscillations in the event of the existence of limit cycles (LC) in 3x3 non-linear systems. It explores the possibility of Stabilising/Quenching the LC by way of signal stabilization using high frequency dither signals both deterministic and random when 3X3 systems exhibit such self-sustained nonlinear oscillations under autonomous state. The present work also explores the suppression limit cycles of 3X3 systems using state feedback by either arbitrary pole placement or optimal selection of pole placement. The complexity involved, in implicit non-memory type nonlinearity for memory type nonlinearities, it is extremely difficult to formulate the problem. Under this circumstance, the harmonic linearization/harmonic balance reduces the complexity considerably. Furthermore, the method is made simpler assuming the whole 3X3 system exhibits the LC predominantly at a single frequency. It is equally a formidable task to make an attempt to suppress the limit cycles for 3X3 systems with memory type nonlinearity in particular. Backlash is one of the nonlinearities commonly occurring in physical systems that limit the performance of speed and position control in robotics, the automation industry, and other occasions of modern applications. The proposed methods are well illustrated through examples and substantiated by digital simulation (a program developed using MATLAB CODES) and the use of the SIMULINK Toolbox of MATLAB software.

: Several fast algebraic methods have been developed for the analysis and synthesis of multivariable control systems. Simple and quick methods based on the block-pulse functions and the matrix continued fractions have been derived for rapid model reduction, system identification, models conversion, rational matrix realization, and design of multivariable control systems. Also, computer-aided methods have been developed for determining the solvents, spectral factors of matrix polynomials and for evaluating the block partial fraction expansions of rational matrices. Furthermore, a quick algorithm based on the matrix sign function has been developed for rapid solution of matrix Riccati equation. The algorithm has successfully been applied to develop a fast suboptimal state-space self-tuner for linear stochastic multivariable systems.

Sensitivity Reduction in Exact Model-Matching of Linear Multivariable Systems

This paper proposes a new design method of robust servo systems for step references and step disturbances. In this method, the tracking characteristics for step references and the feedback characteristics for step disturbances can be determined optimally, using independent quadratic-integral performance indices. Thus, it provides a two-degree-of-freedom design method of control systems in the context of linear-quadratic optimal control. The design procedure consists of two steps. In the first step, a servo system without integral compensation achieving the optimal responses to step references is designed. In the second step, an integral compensator is incorporated, and the feedback gain is determined so as to achieve the optimal responses to step disturbances, without changing the responses to step references. It is shown that as the ‘state weighting matrix’ becomes larger in the second step, the responses to step disturbances become quicker, and that they monotonically tend to the responses of an ‘ideall...

Active feedback control is utilized in this study in order to regulate pen-tip deviations in a novel pen design with the aim to minimize the effects of hand tremors on handwriting. The pen comprises a pendulum-like pen-rod that swings inside a tubular shaped pen casing, and between the pen and the casing, certain compliance and active actuation is considered. Since by the nature of the system dynamics, arbitrary pole placement is not possible in the design of the controller, a nonlinear optimization scheme is constructed to design the controller gains. With these gains, pen-tip deviations are minimized (≈ −47 dB) when the pen casing is subjected tremor-induced cyclic disturbances, and pen-tip response against impulsive perturbations is satisfactorily improved (settling time ≈ 1 sec) while keeping the controller effort around 2 N. Simulation studies are presented comparing the efficacy of the proposed controller with respect to a passively controlled pen, along with trade-offs within the design parameter space.

A positioning system for a direct-drive servo-motor is considered. The controller consists of a PD (proportional plus derivative) or a PID (proportional plus integral plus derivative) feedback regulator to which feedforward compensation has been added. Stability can be ensured by different approaches. One of these approaches consists of positioning the poles inside the unit circle at predetermined positions chosen arbitrarily. Another approach is to determine the upper and lower boundaries of the controller's gains and then to allow the poles to vary freely inside the stability zone without any further action, as long as they remain inside this zone. Possible implementations of a PD and of a PID have been studied. Stability conditions have been determined for each case by arbitrary pole placement as well as by the determination of the upper and lower boundaries on the controller's gains. >

Let G be a strictly proper, rational m × l matrix, with controllability indices λ1≥λ2≥…≥λ l and observability indices μ1≥μ2≥…≥μm. Also let ϕ1 ϕ2…, ϕ l be monic polynomials, where ϕi divides ϕi−1, i = 2, 3,…,l. Does there exist a proper rational feedback matrix K which makes the invariant polynomials of the resulting closed-loop system equal to the ϕi ? What conditions will ensure that there always exists a suitable K ? These questions define the general problem of pole assignment. A number of results are proved in relation to this problem. In particular it is shown that a suitable K always exists if the degrees of the ϕ i satisfy where equality holds when k = l. This contains earlier results of Wonham, Rosenbrock, Brasch and Pearson, and others. A dual result is available in which the roles of the λ i and μ i are interchanged.

This paper proposes an approach of Internal Model Control (IMC) for the linear sampled multivariable non-square systems with outputs more than inputs. The proposed IMC, which is based on a specific inversion principle, is the extension of to the case of square systems, in the controller design, a simple method is presented to add a complementary transfer function matrix of the process to obtain a square transfer matrix. To guarantee the robust stability of the system and disturbance rejection, the controller design procedure is proposed to obtain the wish control. Simulation illustrate the effectiveness of the proposed method for non-square multivariable processes.

The design of multivariable control systems for modern applications is an important challenge to the control system engineer. Active control of metal machining operations, control of gas turbine operations, and chemical process control are current areas of interest. In precision machining operations where tolerances of a few microinches are required, in-process control with several servos will be required. These servos could control spindle axis location and tool position about several axes, thus forming a multi-input/multi-output system. Modern gas turbines, required to operate over more extended regimes, are provided with multiple controls, e.g., nozzle settings and fuel flows which must be implemented in some rational manner. In the chemical process industries, there are many examples of multivariable systems with several control variables and several desired or controlled outputs. One control approach considers a separate system for each of the controlled variables, so that a change in one input will produce an interaction effect that must be managed by another separate system. This approach is attractive and straightforward to implement, but current practice shows significant coupling effects. In order to reduce or eliminate interaction, a control algorithm, with strong integral compensation, for a sixth-order, two-input, two-output linear plant with dynamic coupling is proposed. Decoupling filters are not used. The primary goal is to realize a substantial reduction in the coupling effects when a step input is used for a single variable. A secondary goal is to achieve deadbeat response for the controlled variable to the step input. Moreover, these goals are to be attained in the presence of significant changes in the system parameters or in the presence of arbitrary external disturbances, i.e., robustness is required. The control strategy uses cascaded integral error compensation that permits conceptual division of the network so that two single-input/single-output systems result. Coupling effects are treated as arbitrary disturbances. Poles for each loop are placed on the real axis in the left-half plane. Numerical solutions to the system equations show that this approach produces a system that achieves effective decoupling and robustness simultaneously. It is significantly superior to Proportional-Integral-Derivative controllers which also are considered in detail.

The problem of the specification of the order and structure of a linear dynamic compensator in order to obtain arbitrary pole placement in a closed-loop linear system comprised of the compensator in cascade with a linear plant is discussed. A significant application of the theory is to the design of optimal systems in those cases where not all the state variables of the plant can be measured. These results permit a completely algorithmized approach to the design of compensators for linear systems.

Higher-order continuous-time implicit systems: Consistency and weak consistency, impulse controllability, geometric concepts, and invertibility properties