An important property of non-minimum-phase multiple-input-multiple-output feedback systems

The non-minimum-phase (NMP( property is easily determined from the requirement that the plant input is bounded. In the single-input-single-output (SISO) system, a right-half-plane (RHP) plant zero at s = b constrains the system transfer function to have a zero at b. Also, the available feedback benefits are significantly restricted. The n × n multiple-input-multiple-output (MIMO) system is NMP if the plant determinant δhas any RHP zeros, say at plant transfer matrix and T = [t_ij is the closed-loop system transfer matrix. It has been thought that all n²t_ij (and the n₂ plant disturbance response function rfj), must suffer from the NMP liability in their feedback properties. It is shown that only one row of need so suffer, with a any fixed integer in [1, n].The remaining n(n — 1) elements can be completely free of the NMP liability. A mathematically rigorous synthesis technique previously developed for MP systems is shown to be well suited for precise numerical design for such NMP MIMO plants with significant uncertainties. In this technique, the MIMO design problem is converted into a number of equivalent SISO problems. An example involving disturbance attenuation in a highly uncertain 2×2 NMP plant is included.     

Smith predictor with inverted decoupling for square multivariable time delay systems

This paper presents a new methodology to design multivariable Smith predictor for n×n processes with multiple time delays based on the centralised inverted decoupling structure. The controller elements are calculated in order to achieve good reference tracking and decoupling response. Independent of the system size, very simple general expressions for the controller elements are obtained. The realisability conditions are provided and the particular case of processes with all of its elements as first-order plus time delay systems is discussed in more detail. A diagonal filter is added to the proposed control structure in order to improve the disturbance rejection without modifying the nominal set-point response and to obtain a stable output prediction in unstable plants. The effectiveness of the method is illustrated through different simulation examples in comparison with other works.     

Quantitative synthesis of m × n feedback systems with uncertain plants

There has appeared in the past decade a quantitative feedback synthesis for uncertain multiple-input-multiple-output n × nplants, based on Schauder's fixed point theorem. This paper is devoted to the extension of the quantitative feedback theory to uncertain m × nplants P, m > n. Pis imbedded in a feedback system with compensation operators F, G and provision for command inputs r into the system. There is given a set of acceptable plant outputs. F and G are to be designed so that in response to r, the plant outputs y will be inside the assigned performance bounds, despite the uncertainty in P.

A synthesis method, with considerable transparency, is presented. Available design freedom is used to minimize the cost of feedback which is in the band widths of the loop transfer functions. The designer can see the trade-offs between the loops, and between their bandwidths and the complexity of G, and may choose to sacrifice one Tor the other. The design technique is best suited for ‘basically non-interacting’ performance specifications, wherein output yj(j = 1,2,..., n) is to be primarily due to the rjwith relatively small-components due to the r_k(fc = 1, 2,..., m, k# j). A detailed 3×2 numerical design example is presented.     

Improved Controller Design for Uncertain Positive Systems and its Extension to Uncertain Positive Switched Systems

This paper is concerned with the controller design of uncertain positive systems. First, we decompose the feedback gain matrix K_m×n into m×n non‐negative components and m×n non‐positive components. For the non‐negative components, each component contains only one positive element and the other elements are zero. Similarly, each non‐positive component contains only one negative element and the other ones are zero. Then, a simple and effective controller design approach of uncertain positive systems is proposed by incorporating the decomposed feedback gain matrix into the resulting closed‐loop systems and further applied to uncertain positive switched systems. It is shown that the designed controller is less conservative compared with those in the literature. Finally, a numerical example is provided to verify the validity of the proposed design.     

Optimal diagonal scaling for infinity-norm optimization

A solution is presented for the previously unsolved diagonally scaled multivariable infinity-norm optimization problem of minimizing D(s)(A(s) + Ψ(s) X(s))D⁻¹(s)_∞ over the set of stable minimum-phase diagonal D(s) and stable X(s). This problem is of central importance in the synthesis of feedback control laws for robust stability and insensitivity in the presence of ‘structured’ plant uncertainty. The result facilitates the design of feedback controllers which optimize the ‘excess stability margin’ [3] (or, equivalently, the ‘structured singular value μ’ [4]) of diagonally perturbed feedback systems.     

Inverted decoupling internal model control for square stable multivariable time delay systems

This paper presents a new tuning methodology of the main controller of an internal model control structure for n × n stable multivariable processes with multiple time delays based on the centralized inverted decoupling structure. Independently of the system size, very simple general expressions for the controller elements are obtained. The realizability conditions are provided and the specification of the closed-loop requirements is explained. A diagonal filter is added to the proposed control structure in order to improve the disturbance rejection without modifying the nominal set-point response. The effectiveness of the method is illustrated through different simulation examples in comparison with other works.     

An effective online teaching method: the combination of collaborative learning with initiation and self-regulation learning with feedback

In modern business environments, work and tasks have become more complex and require more interdisciplinary skills to complete, including collaborative and computing skills for website design. However, the computing education in Taiwan can hardly be recognised as effective in developing and transforming students into competitive employees. In this regard, the author adopted collaborative learning (CL) with initiation and self-regulated learning (SRL) with feedback to develop students' collaborative skills and regular learning habits and further contribute to practical computing skills for website design. This study comprised an experiment that included 279 second-year university students from five class sections, including four experimental groups (CISF group, n = 57; CIS group, n = 53; CI group, n = 68; C group, n = 68), and a control group (T group, n = 33). The results reveal that students who received the combined treatment of online CL with initiation and SRL with feedback attained the best grades for their computing skills for website design among the five groups. The author further discusses the implications for teachers, schools and educators who plan to design practical scenarios and online learning activities for their students.     

Multivariable PID control by decoupling

This paper presents a new methodology to design multivariable proportional-integral-derivative (PID) controllers based on decoupling control. The method is presented for general n × n processes. In the design procedure, an ideal decoupling control with integral action is designed to minimise interactions. It depends on the desired open-loop processes that are specified according to realisability conditions and desired closed-loop performance specifications. These realisability conditions are stated and three common cases to define the open-loop processes are studied and proposed. Then, controller elements are approximated to PID structure. From a practical point of view, the wind-up problem is also considered and a new anti-wind-up scheme for multivariable PID controller is proposed. Comparisons with other works demonstrate the effectiveness of the methodology through the use of several simulation examples and an experimental lab process.     

Quantitative non‐diagonal controller design for multivariable systems with uncertainty

The loop coupling reduction of multivariable systems under the presence of plant uncertainty is currently a most discussed topic. Following the ideas suggested by Horowitz, in this paper the role played by the non‐diagonal controller elements is analysed in order to state a design methodology. Thus, the definition of a coupling matrix and a quality function of the non‐diagonal elements come into use to quantify the amount of loop interaction and to design the controllers, respectively. This yields a criterion that makes possible to propose a sequential design methodology of the fully populated matrix controller, in the quantitative feedback theory (QFT) robust control frame. Finally, a real example with the heat exchanger of a pasteurization plant is included to show the practical use of this technique. Copyright © 2002 John Wiley & Sons, Ltd.     

Quantitative non-linear compensation design for saturating unstable uncertain plants

Consider an unstable uncertain plant controlled by an element Np, which saturates when its input |x| ≥ M. The system can be stabilized by means of feedback, which, however, is absent during Np saturation. If the saturation interval is long enough, it is impossible to recover system stability via Np. This paper presents a synthesis technique for ensuring that Np does not saturate despite very large command inputs. The basic idea is to prevent |x|>M, via an added saturating element N with saturation level m, which in turn is determined by |x|. A systematic, quantitative design technique is presented for unstable plants with large uncertainty, to achieve (a) desired performance tolerances over the linear range (small command inputs) and (b)acceptable but unavoidably slower response for large command inputs. Both (a) and (b) are achieved over the specified extent of plant uncertainty. The design technique makes use of several previously developed quantitative synthesis theories for minimum- and non-minimum-phase uncertain plants in linear operation, and for uncertain minimum-phase stable plants subject to saturation.     

Iterative solution of a class of linear equations with application to reconstruction of three-dimensional object arrays†

An iterative algorithm baaed on probabilistic estimation is described for obtaining the minimum-norm solution of a very large, consistent, linear system of equations AX = g where A is an (m × n) matrix with non-negative elements, x and g are respectively (n × 1) and (m × 1) vectors with positive components.

This algorithm will find application in the reconstruction of three-dimensional object arrays from projections and in several other areas.     

On the robustness of multivariable linear feedback systems

One of the useful indicators of the robustness of a multivariable linear feedback system is the largest singular value of the nominal closed-loop transfer matrix. It is shown that while comparing different, not necessarily diagonal, closed-loop transfer matrices which have the same diagonal elements, the diagonal closed-loop transfer matrix has the greatest robustness. For plants with not "too" large parameter uncertainty, this result also guarantees the maximization of disturbance rejection, and the minimization of the control signal "power" at the plant's input.     

Iterative identification and two step control design for partially unknown unstable plants

In this paper we shall extend the applications of iterative identification and control design to partially unknown unstable plants. We show that by employong a two step approach, where an unstable plant is first stabilized by a parallel feedback stabilizer, it is possible to design systematically an overall closed-loop system that has good step responses with little overshoots by using the iterative identification and control design methodology. Furthermore, this approach easily preserves the simplicity of an IMC design through tuning the overall designed closed-loop bandwidth with a single design parameter. Specifically, similarly to situations where the plant is stable (apart from possibly including a simple integrator), we can design a system with a small initial overall designed closed-loop bandwidth (after the plant is stabilized by a known parallel feedback stabilizer) such that high frequency unmodelled dynamics of the plant are not overly excited. Through iterative applications of a control-relevant closed-loop system identification procedure and the standard IMC design method to the stabilized plant, the overall designed closed-loop bandwidth of the system can be widened progressively while maintaining good step responses with little overshoots. Two simulation examples are employed to illustrate the method. These examples show that, irrespective of the presence of adverse unstable real pole-zero structures, the expected results are achievable by the method described.     

QFT design for uncertain non-minimum phase and unstable plants revisited

Design method for uncertain non-minimum phase and unstable plants in the quantitative feedback theory (QFT) developed by Horowitz and Sidi is revisited in this paper. It is illustrated that the existing method may not work since some design rules have not been clearly specified by several examples including non-minimum phase plants and unstable plants. Then stability of a new nominal plant is carefully examined and analysed, and an improved design method is presented. The result in this paper provides mathematical justification of the QFT design procedure for nonminimum phase and unstable plants in Horowitz and Sidi (1978) and Horowitz (1992).     

Matrix WA + AtW and its applications

For matrix A, with off–diagonal elements not necessarily of the same sign, conditions are obtained for the existence of a positive definite diagonal matrix W, such that matrix WA + A^tW is positive definite; and the applications of the latter to the determination of the stability of interval matrices are considered.     

Use of the design freedom of time-optimal control

A method for synthesizing a state-feedback time-optimal controller for linear n-state, r-input discrete-time systems is developed. The nonuniqueness of the solution is organized in such a way as to select the controller with the minimum value of the Frobenius norm of the feedback gain matrix K. It is shown that among the rn design parameters in the parametric form of K only r(n − rp_r) need to be considered in the search technique, where p_r is the order of the smallest Jordan block in the Jordan form of the closed-loop system.     

Linear systems with bounded inputs: global stabilization with eigenvalue placement

This work presents a technique for obtaining a bounded continuous feedback control function which stabilizes a linear system in a certain region. If the open-loop system has no eigenvalues with positive real part, the region of attraction of the resulting closed-loop system is all ℝⁿ, i.e., the feedback control is a global stabilizer; otherwise, the region contains an invariant (‘cylindric-like’) set where the controller does not saturate. The proposed control is a linear-like feedback control with state-dependent gains. The gains become implicitly defined in terms of a nonlinear scalar equation. The control function coincides in an ellipsoidal neighbourhood of the origin with a linear feedback law which is a solution of a linear quadratic regulator problem. This design allows eigenvalue placement in a specified region. © 1997 by John Wiley & Sons, Ltd.     

Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation

This paper mainly discusses fractional differential approach to detecting textural features of digital image and its fractional differential filter. Firstly, both the geo- metric meaning and the kinetic physical meaning of fractional differential are clearly explained in view of information theory and kinetics, respectively. Secondly, it puts forward and discusses the definitions and theories of fractional stationary point, fractional equilibrium coefficient, fractional stable coefficient, and fractional grayscale co-occurrence matrix. At the same time, it particularly discusses frac- tional grayscale co-occurrence matrix approach to detecting textural features of digital image. Thirdly, it discusses in detail the structures and parameters of nxn any order fractional differential mask on negative x-coordinate, positive x-coordi- nate, negative y-coordinate, positive y-coordinate, left downward diagonal, left upward diagonal, right downward diagonal, and right upward diagonal, respectively. Furthermore, it discusses the numerical implementation algorithms of fractional differential mask for digital image. Lastly, based on the above-mentioned discus- sion, it puts forward and discusses the theory and implementation of fractional differential filter for digital image. Experiments show that the fractional differential-based image operator has excellent feedback for enhancing the textural details of rich-grained digital images.     

Generation of stable limit cycles in controllable linear systems

In this paper we prove that any controllable linear systems , admits a polynomial feedback u= u(x) such that the closed-loop system admits an orbitally asymptotically stable limit cycle.Moreover, we prove that for any positive integer n, there exists an nth-order polynomial, autonomous, ordinary differential equation with a unique limit cycle.