Making a polynomial Hurwitz-invariant by choice of feedback gains†

An integrated design procedure is developed for a modified Smith predictor and associated controller for linear time-delay systems having transfer functions of the form k ₁, A exp (—sT)/B, where A and B are monic polynomials in s of degree n — l and n, respectively. A is Hurwitz and B has a single right-half-plane root at s = λ. For l=1,2,3, an augmented PI controller guarantees asymptotic stability for λT less than an l-dependent limit. The procedure for l = 3 is extended to l = 4 with the introduction of derivative action into the controller. Design arguments are on root locus topology, and on Nyquist analysis applied to an auxiliary system.     

A new parameterization of stable polynomials

In this note, we develop a new characterization of stable polynomials. Specifically, given n positive, ordered numbers (frequencies), we develop a procedure for constructing a stable degree n monic polynomial with real coefficients. This construction can be viewed as a mapping from the space of n ordered frequencies to the space of stable degree n monic polynomials. The mapping is one-one and onto, thereby giving a complete parameterization of all stable, degree n monic polynomials. We show how the result can be used to generate parameterizations of stabilizing fixed-order proper controllers for unity feedback systems. We apply these results in the development of stability margin lower bounds for systems with parameter uncertainty.     

基于改进Smith预估控制结构的二自由度PID控制

针对工业过程中的二阶不稳定时滞过程, 基于改进史密斯预估控制结构提出了一种简单的两自由度控制方案.设定值跟踪控制器和扰动抑制控制器采用同一设计程序, 并基于内模控制原理提出了控制器解析设计方案.设定值跟踪控制器和抗扰动控制器可分别通过单性能参数独立调节和优化, 每个控制器都具有PID形式, 给出了控制器调整参数的选择范围和扰动抑制闭环保证鲁棒稳定性的条件.仿真实例验证了提出方法对于近期其他方法的优越性.     

Output-feedback sampled-data polynomial controller for nonlinear systems

This paper presents the stability analysis and control synthesis for a sampled-data control system which consists of a nonlinear plant and an output-feedback sampled-data polynomial controller connected in a closed loop. The output-feedback sampled-data polynomial controller, which can be implemented by a microcontroller or a digital computer, is proposed to stabilize the nonlinear plant. Based on the Lyapunov stability theory, stability conditions in terms of sum of squares are obtained to guarantee the stability and to aid the design of a polynomial controller. A simulation example is given to demonstrate the effectiveness of the proposed control approach.     

Robust pole assignment via reflection coefficients of polynomials

A robust version of the output controller design for discrete-time systems is introduced. Instead of a single stable point a stable polytope (or simplex) is preselected in the closed-loop characteristic polynomial coefficients space. A constructive procedure for generating stable polytopes is given starting from the unit hypercube of reflection coefficients of monic polynomials. This procedure is quite straightforward because for a special family of polynomials the linear cover of so-called reflection vectors is stable. The roots placement of reflection vectors is studied. If a stable target simplex is preselected then the robust output controller design task is solved by quadratic programming approach.     

Properties of the entire set of Hurwitz polynomials and stabilityanalysis of polynomial families

It is proved in this paper that all Hurwitz polynomials of order not less than n form two simply connected Borel cones in the polynomial parameter space. Based on this result, edge theorems for Hurwitz stability of general polyhedrons of polynomials and boundary theorems for Hurwitz stability of compact sets of polynomials are obtained. Both cases of families of polynomials with dependent and independent coefficients are considered. Different from the previous ones, our edge theorems and boundary theorems are applicable to both monic and nonmonic polynomial families and do not require the convexity or the connectivity of the set of polynomials. Moreover, our boundary theorem for families of polynomials with dependent coefficients does not require the coefficient dependency relation to be affine     

Design of digital PID controllers using the parameter space approach

In this paper, a parameter space approach is taken for designing digital PID controllers. The stability domains of the coefficients of the controllers are computed. The existing continuous-time results are extended to the case of discrete-time systems. In this approach, the stability region is obtained in the plane of two auxiliary controller coefficients by assuming a fixed value for a third auxiliary controller coefficient. The stability region is defined by several line segments or equivalently by several linear equalities and inequalities. Then, through mapping from the auxiliary coefficient space to the original controller coefficient space, exact stability domain in the (K_P ???K_I ???K_D ) space is obtained. The method is also extended for locating the closed-loop poles of PID control systems inside the circles with arbitrary radii, centred at the origin of the z-plane. The results can be used in the design of dead-beat control systems.     

一类状态及输入均有区间变时滞的线性系统之镇定

对一类同时具有状态和输入变时滞的线性系统(x)(t)=Ax(t) A1x(t-τ1(t)) B1u(t) B2υ(t-τ2(t)),利用一种时滞划分形式的Lyapunov-Krasovskii泛函讨论其镇定问题.得到通过状态反馈控制使闭环系统稳定的时滞相关的镇定准则.对(A,B1)和(A A1,B1)都不可控的数值实例,利用本文给出的准则都能得到有效的控制.     

时滞过程改进型Smith预估器的整定

证明Majhi和Atherton(1999)文所提出的改进型Smith预估器等价于一改进的内模控制结构 (IMC), 并对该结构提出一种三阶段设计方法. 为获得扰动抑制和稳定鲁棒性的均衡, 采用了鲁棒控制方法来整定反馈环控制器. 针对某些典型的积分和不稳定时滞过程的设计表明所提方法能获得较好的扰动抑制和稳定鲁棒性的均衡.     

Unconstrained H∞ predictive control with H∞ prediction: single‐input–single‐output case

Does the replacement of the quadratic (H₂) predictor by the worst‐case (H_∞, or cumulative minimax) predictor robustify the predictive control laws? The present work provides a partial answer to this question, positive for the examples considered, representative of three broad classes of systems. The H_∞ prediction is demonstrated to be a powerful and convenient tool for frequency shaping of the gain of the closed‐loop complementary sensitivity function, capable of robustifying the closed loop for systems with different stability properties. The H_∞‐optimal k‐step ahead predictor is derived for an unstable single‐input–single‐ output CARMA model. A BIBO unstable filter for the disturbance rejection is obtained using the internal model principle and included into the closed loop, and the H_∞ predictor is applied to the combination of this filter with the plant. The sum over a finite horizon of the current and the predicted tracking error and control signal power spectral densities (PSDs) is decomposed into two parts, one induced by the worst‐case predicted disturbance and the other—by the known future reference input. A two degrees of freedom algorithm, referred to as the multi‐step closed‐loop polynomial H_∞ predictive control law, is obtained that minimizes the peaks of the PSD of the first part and the integral on the unit circle of the PSD of the second. It is demonstrated on several systems that H_∞ prediction introduces a very intuitive tuning knob in the form of the prediction horizon capable of setting a trade‐off between the steady‐state disturbance rejection perfor mance in terms of the output error variance and the closed‐loop robustness, however the efficacy of the knob strongly depends on the stability properties of the system and its inverse. The trade‐off becomes less pronounced or completely disappears when the H_∞ predictor is replaced by the quadratic one. Copyright © 2000 John Wiley & Sons, Ltd.     

On Polynomial Decompositions

We present a new polynomial decomposition which generalizes the functional and homogeneous bivariate decomposition of irreducible monic polynomials in one variable over the rationals. With these decompositions it is possible to calculate the roots of an imprimitive polynomial by solving polynomial equations of lower degree.     

Galois Action on Solutions of a Differential Equation

Consider a second-order differential equation of the form y″ + ay ′ + by = 0 with a, b ϵ Q(x). Kovacic's algorithm tries to compute a solution of the associated Riccati equation that is algebraic and of minimal degree over Q̄ (x). The coefficients of the monic irreducible polynomial of this solution are in C(x), where C is a finite algebraic extension of Q. In this paper we give a bound for the degree of the extension C ⊃ Q. Similar results are obtained for third-order differential equations.     

Computation of stabilizing PI-PD controllers

The PI-PD controller structure provides an excellent four-parameter controller for control of integrating, unstable and resonant processes to set point changes while the conventional PID controller has limitations in controlling such systems. In this paper, a graphical method for the computation of all stabilizing PI-PD controllers is given. The proposed method is based on plotting the stability boundary locus, which is a locus dependent on the parameters of the controller and frequency, in the parameter plane. The stability boundary loci are first obtained in the (K _d , K _f ) and (K _p , K _i ) planes and then it is shown that all the stabilizing values of the parameters of a PI-PD controller can be found. Computation of stabilizing PI-PD controllers which achieve user specified gain and phase margins is also studied. The method is used to design robust PI-PD controllers for control systems with parametric uncertainties. A design procedure for interval control systems is proposed. Examples are given to show the benefit of the method presented. Recommended by Editorial Board member Jietae Lee under the direction of Editor Young Il Lee. Nusret Tan was born in Malatya, Turkey, in 1971. He received his B.Sc. degree in Electrical and Electronics Engineering from Hacettepe University, Ankara, Turkey, in 1994. He received the Ph.D. degree in Control Engineering from University of Sussex, Brighton, U.K., in 2000. He is currently working as an Associate Professor in the Department of Electrical and Electronics Engineering at Inonu University, Malatya, Turkey. His primary research interest lies in the area of systems and control.     

Optimally robust H∞ polynomial fuzzy controller design using quantum-inspired evolutionary algorithm

This paper proposes an optimally robust H_∞ polynomial fuzzy controller design using quantum-inspired evolutionary algorithm (QEA) for continuous/discrete time polynomial fuzzy systems with model uncertainties and external disturbances. To improve control performance, QEA is adopted to evolve optimal control gains with a fitness function that is defined by performance requirements. The stability and robustness of the control system are then guaranteed by the proposed robust H_∞ stability conditions, which are formed by the sum of squares (SOS) method. By using the principle of copositivity, novel relaxed SOS-based stability conditions are derived to reduce the conservativeness of solving SOS-based stability conditions, while the feasible solution space is broadened. Four numerical examples demonstrate the effectiveness of the proposed approaches.     

Instability analysis and improvement of robustness of adaptive control

The effects of unmodeled high frequency dynamics and bounded disturbances on stability and performance of adaptive control schemes are analyzed. Five possible types of instability mechanisms—parameter drift, ‘linear’ instability, ‘fast adaptation’ instability, ‘high frequency’ instability, and ‘throughput’ instability—are analyzed using simple examples. A procedure is used to construct Lyapunov-like functions for a modified adaptive controller applied to a dominant plant of relative degree one, in the presence of parasitics and disturbances, and obtain sufficient conditions under which none of the five types of instability can occur. The modified scheme is robust in the sense that it guarantees the existence of a large region of attraction from which all the trajectories remain bounded and the state errors converge exponentially to a much smaller residual set. The size of the region of attraction depends on the speed of parasitics in such a way that as the parasitics become infinitely fast, the region of attraction becomes the whole space.     

Algorithms for polynomials in Bernstein form

Aspects of the formulation and numerical stability of geometric modeling algorithms in the Bernstein polynomial basis are further explored. Bernstein forms for various basic polynomial procedures (the arithmetic operations, substitution of polynomials, elimination of variables, determination of greatest common divisors, etc.) required in such algorithms are developed, and are found to be of similar complexity to their customary power forms. This establishes the viability of systematic computation with the Bernstein form, avoiding the need for (numerically unstable) basis conversions.Procedures which nominally improve root condition numbers—the power-to-Bernstein conversion and Bernstein degree elevation and subdivision techniques—are examined in greater detail. These procedures are found to have a relatively poor inherent conditioning, which essentially cancels the expected improvement, and their explicit use does not therefore provide a means of enhancing numerical stability.We also examine the condition of computation in floating point arithmetic of the power and Bernstein formulations for the basic polynomial procedures. However, the detailed dependence of the computational errors on the input parameters, the algorithm structure, and the floating point system preclude comparisons as definitive and general as those pertaining to the root condition numbers in the power and Bernstein bases. Empirical evaluation of carefully selected test cases may provide firmer conclusions in this regard.     

Composite State Feedback of the Finite Frequency H∞ Control for Discrete‐Time Singularly Perturbed Systems

This paper mainly is concerned with the finite frequency H_∞ control for the discrete‐time singularly perturbed systems. A state feedback controller is designed to stabilize the whole system and to satisfy the desired design specifications. The generalized Kalman–Yakubovich–Popov (GKYP) lemma is used to convert the related frequency domain inequalities in finite frequency ranges to feasible linear matrix inequalities. Based on the Lyapunov stability method, stable conditions are obtained for discrete‐time singularly perturbed systems. A bounded real lemma then is derived, which characterizes the H_∞ norm performance in specific frequency ranges. Furthermore, the approach for the design of a composite state feedback controller is put forward combined with the unique frequency characteristics of singularly perturbed systems. Detailed analysis of the performance achieved by the piecewise composite controller is provided when it is applied to the original system, and the effectiveness and merits of the proposed controller are illustrated with a numerical result.     

Lyapunov-based exact stability analysis and synthesis for linear single-parameter dependent systems

We propose a class of polynomially parameter-dependent quadratic (PPDQ) Lyapunov functions for assessing the stability of single-parameter-dependent linear, time-invariant, (s-PDLTI) systems in a non-conservative manner. It is shown that the stability of s-PDLTI systems is equivalent to the existence of a PPDQ Lyapunov function. A bound on the degree of the polynomial dependence of the Lyapunov function in the system parameter is given explicitly. The resulting stability conditions are expressed in terms of a set of matrix inequalities whose feasibility over a compact and connected set can be cast as a convex, finite-dimensional optimisation problem. Extensions of the approach to state-feedback controller synthesis are also provided.     

Globally convergent adaptive pole assignment for a class of non-minimum-phase stochastic systems

A key difficulty of the explicit approach to self-tuning control—both theoretically and computationally—is the need to solve a polynomial identity to generate the required controller coefficients. For systems with uncorrelated output noise, however, the identity has a simple solution, and in this paper the implications of this phenomenon are discussed in relation to self-tuning regulation. A suitable explicit algorithm is introduced, and it is shown that, under certain conditions, global stability and system identifiability can be established without recourse to sophisticated estimator management techniques.