Compensation of uncertain continuous systems by using the internal model control principle

In this paper the design of compensators for uncertain continuous plants is investigated. The standard derived compensators are based on the application of the internal model control (IMC) method. The required a priori knowledge on the plant is rather weak, namely, an upper bound of the plant relative order, the numbers of the strictly unstable and critically unstable plant poles being integrators and upper and lower bounds of the amplitude-versus-frequency plot over the low frequency band in the case of minimum-phase open-loop systems. If the open-loop system has unstable zeros and/or poles then the above bounds are required to be known for a modified magnitude plot which substitutes the unstable zeros (poles) by stable poles (zeros) which are their complex-conjugate reflections on the left-hand plane. An absolute upper bound of the open-loop phase plot is obtained on a finite frequency interval which allows the closed-loop system to guarantee a prescribed relative stability in many practical situations. The method is dependent on the alternative design of phase lead/lag classical compensators and to indirect adaptive control situations where the adaptive identifier is used for the parametrization of the adaptive controller.     

Feedback limitations of linear sampled‐data periodic digital control

We present a set of feedback limitations for linear time‐invariant systems controlled by periodic digital controllers based upon an analysis of the inter‐sample response of the closed‐loop system to sinusoidal inputs. Fundamental sensitivity and complementary sensitivity functions govern the fundamental and harmonic components of the continuous closed‐loop response. The continuous and discrete response of the system is sensitive to variations in the analog plant at frequencies integer multiples of ω_s/N away from the excitation frequency, where ω_s is the sampling frequency and N is the period of the controller. These functions satisfy interpolation and integral constraints due to open‐loop non‐minimum phase zeros and unstable poles. In addition, the use of periodic digital control may result in a reduction in closed‐loop bandwidth. Copyright © 2000 John Wiley & Sons, Ltd.     

Minimization of the Bode sensitivity integral

In this paper we present two new results on Bode-type integrals. Firstly, we obtain, for a given scalar or multivariable continuous-time plant, the infimum of the Bode sensitivity integral which can be obtained with any stabilizing controller. The result involves the unstable plant poles and, perhaps surprisingly, a subset of the plant nonminimum phase zeros. Secondly, we obtain an apparently new expression for the Bode integral for the complementary sensitivity for a stable discrete-time scalar system.     

Controller design for nonlinear systems using the Contoured Robust Controller Bode plot

In this paper, we develop the Contoured Robust Controller Bode (CRCBode) plot and demonstrate its use in the design of robust controllers for nonlinear single‐input single‐output (SISO) systems. The CRCBode plot shows contours (level sets) of a robust performance quantity on the Bode magnitude and phase plots of the controller. An iterative frequency domain loop‐shaping design approach is employed to eliminate all intersections of the controller frequency response with certain ‘forbidden regions,’ indicating that a standard SISO robust stability and performance criterion is satisfied. Nonlinearities are accounted for by avoiding the maximum forbidden regions over a structured uncertainty set consisting of linearizations of the system dynamics about several operating points. We demonstrate this technique by designing and experimentally verifying a flow‐rate controller for a butterfly‐valve based liquid cooling system, which is robust to valve nonlinearities and flow disturbances. Finally, we compare this compensator with one generated using an automated H _∞ synthesis algorithm and discuss the advantages of the CRCBode approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Coupled‐line band‐pass filter with T‐shaped structure for high frequency selectivity and stopband rejection

A coupled‐line band‐pass filter (BPF) with T‐shaped stub structure is presented. Five transmission poles within the passband and eight deep transmission zeros (TZs) from 0 to 2f₀ (f₀ denotes filter's center frequency) are realized through input impedance calculations. With the simple T‐shaped structure, the positions of six TZs can be appropriately adjusted to achieve high frequency selectivity and stopband rejection. For demonstration, a BPF prototype centered at 2.05 GHz is designed and fabricated, whose measured rejection levels are of over 45.5 dB at lower stopband and better than 19.5 dB at upper stopband. The simulation and measurement results are in good agreement, which validates the design idea.     

Fuzzy frequency response: Proposal and application for uncertain dynamic systems

Fuzzy frequency response: proposal and application for uncertain dynamic systems is proposed in this paper. The uncertain dynamic system is partitioned into several linear sub-models, in terms of transfer function, and organized into Takagi–Sugeno (TS) fuzzy structure. The main contribution of this approach is demonstrated, from a Theorem, that the fuzzy frequency response is a bound in the magnitude and phase Bode plots. The analysis, from the fuzzy dynamic model, at low and high frequencies, is obtained varying the frequency ω from zero to infinity.     

H∞-optimal robust regulation of MIMO systems

This paper introduces a new synthesis method, based on a computation method for solving H^∞-optimization problem given by Glover (1986) and a technique ofjω-axis shifting, for designing an H^∞-optimal robust controller for the MIMO system. With this method we can optimize the excess stability margin under the requirement that all the poles of the nominal closed-loop system (Cl-system) be located left of the jω-axis with a required distance d and ensure asymptotic regulation and disturbance rejection for a class of system input and disturbance if the plant perturbation does not exceed the excess stability margin. Another advantage is that there are not any constraints on the poles and zeros of the plant when using this method     

Pole‐zero analysis and wavelength scaling of carbon nanotube antennas

Carbon nanotube (CN) antennas have applications in the THz electromagnetic spectrum. Nanotubes have a highly dispersive and frequency dependent conductivity model. In this article, we compare the poles and zeros in the input impedance of CN antennas at different lengths. We used model‐based parameter estimation to approximate the input impedance of the antenna with a rational function in the complex frequency domain. Despite dispersive conductivity of CN, the imaginary part of the poles and zeros are respectively the integer multiples and odd multiples of the imaginary part of the first pole and zero. However, the real part of poles is almost constant, while the pattern was not observed for the real part of zeros. We also show that CN dipoles operating between 43 and 53 GHz are well matched if the source impedance is much higher than 50 ohms, and even higher than 12.9 kΩ. The fundamental resonances (f₀) of CN dipoles plotted versus their inverse‐half‐length (1/L) are linearly related, but the intercept of the fitted straight line is non‐zero unlike that for perfect electric conductor (PEC) dipoles. This leads to non‐linear variation in wavelength scaling of CN dipoles. The resonant CN antennas are relatively much shorter than PEC dipoles.     

Integral constraints and performance limits on complementary sensitivity: discrete-time systems

We study performance limitation issues for multivariable discrete-time feedback systems. The complementary sensitivity function is employed as a performance measure, and Bode and Poisson-type integral inequalities and -type performance limits are derived. The results exhibit frequency-dependent constraints as well as best achievable limits on the complementary sensitivity function, which are shown to be determined by nonminimum phase zeros, unstable poles, and time delays. In particular, the directions of such zeros and poles are seen to play a central role to this effect.     

On the convergence order of a modified method for simultaneous finding polynomial zeros

Using Newton's corrections and Gauss-Seidel approach, a modification of single-step method [1] for the simultaneous finding all zeros of ann-th degree polynomial is formulated in this paper. It is shown thatR-order of convergence of the presented method is at least 2(1+τ _n ) where τ _n ∈(1,2) is the unique positive zero of the polynomial \(\tilde f_n (\tau ) = \tau ^n - \tau - 1\) . Faster convergence of the modified method in reference to the similar methods is attained without additional calculations. Comparison is performed in the example of an algebraic equation.     

The asymptotic behaviour,the angles of departure,and the angles of approach,of the characteristic frequency loci

In algebraic function theory, there is a well established method which uses ‘Newton's diagram’ to find the series expansions of an algebraic function q(x) in the neighbourhood of a point x₀ . In this paper it is shown how, for a linear, time-invariant, multi-variable feedback system, this method can be used to find :

(i) the asymptotic behaviour of the characteristic frequency loci (multivariable root loci) ;

(ii) the angles of departure of the characteristic frequency loci from the open-loop poles ; and

(iii) the angles of approach of the characteristic frequency loci to the finite zeros of such a system.     

A new approach for the design of multivariable feedback systems

A new design technique for multivariable feedback systems is presented. In this approach, n —1 open-loop transfer functions at different inputs of the plant, with all other feedback paths closed, are specified in advance, and are achieved exactly. The nth open-loop transfer function is a by-product of the design process, such that the overall feedback system is stabilized. The design approach is fitted to solve problems in which the plant elements can have non-stable poles and non-minimum phase zeros. The design process is straightforward, no iterations are necessary, and the achieved design copes exactly with the design specifications. The gainbandwidths of the different lis and the overall loop gain l* might be constrained due to non-stable poles and zeros of the plant elements. Based on the obtained different loop gains, any input output matrix T can bo achieved with the aid of an appropriate prefilter matrix F.     

Logarithmic integrals, interpolation bounds, and performancelimitations in MIMO feedback systems

We study performance limitation issues found in linear multivariable feedback systems. Our main contributions include Bode and Poisson type integral inequalities and performance limits for the sensitivity and complementary sensitivity functions. These results characterize and quantify explicitly how open-loop unstable poles and nonminimum phase zeros may impose inherent limitations on feedback design and fundamental limits on the best achievable performance. The role of time delay is also studied in this context. Most notably, we show that the performance and design limitations in multivariable systems intrinsically depend on the locations as well as directions of unstable poles and nonminimum phase zeros, and in particular, on how pole and zero directions are aligned. The latter is characterized by angles measuring the mutual orientation between zero and pole directions, and it is shown to play a crucial role in multivariable system design     

Parametric system identification on logarithmic frequency responsedata

Gradient search methods that fit the parameters of a user-defined transfer function to experimental logarithmic frequency response data are presented. The methods match a model based on physically significant parameters, including natural frequencies of poles and zeros and damping ratios of complex poles and zeros. The algorithms construct and utilize their own analytical gradient descent functions, based on the desired model. One method attempts to fit both log magnitude and phase, while another identifies a minimum phase transfer function model from only log magnitude frequency response data. The log magnitude algorithm is shown to be superior to traditional methods using nonlogarithmic frequency response data, including those used in commercially available frequency response analyzers. The algorithms are shown to perform well, especially for systems with lightly-damped dynamics     

Design of dominant-type control systems with large parameter variations†

This paper presents a design procedure for dominant-type systems with large plant parameter variations. The principal contribution is that a fourth-order approximation is used in the dominant region instead of a third-order, which up to now had been the most advanced method. The s-domain specifications of the system are assumed to be in the form of an acceptable dominant closed-loop pole region with bounds on the location of the ‘far-off’ closed-loop poles. The design philosophy is to place compensation zeros within the acceptable dominant closed-loop pole region such that the dominant closed-loop poles remain within their prescribed region despite the large variation in the plant parameters. The design procedure is for plants with simultaneous independent variation in the gain factor and a pair of poles. The design is such as to minimize the sensitivity of the system to internal noise.     

On the design,robustness, implementation and use of quasi-linear feedback compensators

A quasi-linear feedback compensator is one in which its poles depend in an appropriate way on its gain. The reason for introducing this new concept was the desire to remove the limitation to performance imposed by a plant with more than one pole in excess of its zeros. In this article it is shown that this objective is realized for plants with zeros in the left half of the complex plane. The consequences are surprising. In time domain it is possible to track arbitrarily fast a class of reference inputs despite a large class of disturbances and uncertainty in plant parameters. The response is non-oscillatory for high enough compensator gains, which is explained by the automatic adaptation of the closed loop poles to stability and stability margins for such gains. And in frequency domain the phase margin tends to 90° while the gain margin and crossover frequency become unlimited.

Technically the design procedure of quasi-linear compensators presented here is based on our theoretical result concerning the asymptotic behaviour of the roots of certain polynomials in a complex variable which depend also on a large positive parameter.

We also show how to implement such quasi-linear compensators in practical feedback control schemes, and their use at lower gains which is the case of most industrial applications.     

PERFORMANCE FUNCTIONS AND SUBLEVEL SETS FOR FREQUENCY DOMAIN DESIGN OF SINGLE LOOP FEEDBACK CONTROL SYSTEMS WITH PLANT UNCERTAINTY

The frequency domain design of control systems involves the fitting of a designable complex function of frequency (e.g., loop transfer function, compensator transfer function) to specification derived constraints in the design space ℂ×ℝ. In the classical Bode approach and its recent modifications the designable function is the loop transfer function, the specifications lead to constraints on the magnitude of the loop transfer function and the constraints have circular cross-sections in ℂ. In more recent design procedures (e.g., H _∞ optimization or QFT) the specifications lead to more complex constraint surfaces in ℂ×ℝ and the design procedure must fit the designable function to these constraints. In this paper we develop explicit representations of some important ℂ×ℝ constraint surfaces encountered in the design of SISO control systems with both non-parametric (unstructured) and parametric plant uncertainty and study the characteristics of their frequency axis cross-sections or level sets: important information use in the fitting process. The results are presented in two systems of co-ordinates (i.e., designable functions): nominal closed loop transfer function T_o(jω) and and nominal open loop transfer function L_o(jω). While the results in T_o co-ordinates are more useful when using mathematical optimization (e.g., H _∞ optimization techniques), the results in L_o co-ordinates have significant advantages of insight and ease of graphical manipulation as demonstrated in QFT. The inclusion of results in both sets of co-ordinates increases their utility for workers in both areas and also reveals some links between these two different approaches. © 1997 by John Wiley & Sons, Ltd.     

Complete solution of the 4‐block H∞ control problem with infinite and finite jω‐axis zeros

This paper discusses the 4‐block H^∞ control problem with infinite and finite jω‐axis invariant zeros in the state‐space realizations of the transfer functions from the control input to the controlled output and from the disturbance input to the measurement output, where these realizations are induced from a stabilizable and detectable realization of the generalized plant. This paper extends the DGKF approach to the H^∞ control problem but permitting infinite and finite jω‐axis invariant zeros by using the eigenstructures related to these zeros. Necessary and sufficient conditions are presented for checking solvability through checking the stabilizing solutions of two reduced‐order Riccati equations and examining matrix norm conditions related to the jω‐axis zeros. The parameterization of all suitable controllers is given in terms of a linear fractional transformation involving a certain fixed transfer function matrix and together with a stable transfer function matrix with gain less than 1 which is free apart from satisfying certain interpolation conditions. Copyright © 2000 John Wiley & Sons, Ltd.     

An Approximation Algorithm for the Fault Tolerant Metric Facility Location Problem

We consider a fault tolerant version of the metric facility location problem in which every city, j, is required to be connected to r _j facilities. We give the first non-trivial approximation algorithm for this problem, having an approximation guarantee of 3 · H _k , where k is the maximum requirement and H _k is the kth harmonic number. Our algorithm is along the lines of [2] for the generalized Steiner network problem. It runs in phases, and each phase, using a generalization of the primal–dual algorithm of [5] for the metric facility location problem, reduces the maximum residual requirement by one.     

An Approximation Algorithm for the Fault Tolerant Metric Facility Location Problem

We consider a fault tolerant version of the metric facility location problem in which every city, j, is required to be connected to r _j facilities. We give the first non-trivial approximation algorithm for this problem, having an approximation guarantee of 3 · H _k , where k is the maximum requirement and H _k is the kth harmonic number. Our algorithm is along the lines of [2] for the generalized Steiner network problem. It runs in phases, and each phase, using a generalization of the primal–dual algorithm of [5] for the metric facility location problem, reduces the maximum residual requirement by one.