Periodic compensation of a class of decentralized systems with fixed modes

This paper first finds that the high frequency (ω) periodic controller suggested in Lee, Meerkov, and Runolfsson (1987) for fixed mode removal, and pole and zero placement, of a class of decentralized systems (A,B_i,C_i) all the strongly interconnected channels of which satisfy C_iB_i=0, causes large, O(ω), oscillations in the outputs. Next it proposes an alternative controller, a dual of the above, that achieves the compensation with oscillations of O(1) only.     

Gleichheit von Produkt und Formalprodukt bei Intervallpolynomen

The paper gives necessary and sufficient conditions for the product of two interval polynomials being equal to their formally defined product. For this purpose criteria must be deduced for which intervalsA _i, B _j the equality(A ₁+...+A _m ) (B ₁ +...+B _n )=A ₁ B ₁ +A ₁ B ₂ +...+A _m B _n (relation of distributivity) holds.     

Construction of superoptimal approximants

LetG be a matrix function of type m×n and suppose thatG is expressible as the sum of anH ^∞ function and a continuous function on the unit circle. Then it is known that there is a unique superoptimal approximant toG inH ^∞: that is, there is a unique analytic matrix functionQ in the open unit disc which minimizess ^∞(G?Q) or, in other words, which minimizes the sequence $$(s_0^\infty (G - Q),s_1^\infty (G - Q),s_2^\infty (G - Q), \ldots )$$ with respect to the lexicographic ordering, wheres _j ^∞ (F)=sup _x∈ s _j (F(z)) ands _j (·) denotes thejth singular value of a matrix. We give a function-theoretic (frequency domain) algorithm for the construction of this approximant. We calculate an example to illustrate the algorithm. The construction works for rationalG, but is also valid for non-rational functions. It is based on the authors' uniqueness proof in [PY1], but contains extra ingredients required to render it practicable, notably one which obviates the need for the preliminary solution of a Nehari problem. We also establish a formula forQ in terms of the maximizing vectors of a sequence of Hankel-type operators.     

Necessary and sufficient conditions for existence of J-spectral factorization for para-Hermitian rational matrix functions

For a para-Hermitian rational matrix function G(λ)=J+C(λI_p−A)⁻¹B, where J=J∗ is invertible, and which has no poles and zeros on the imaginary line, we give necessary and sufficient conditions in terms of A,B,C and J for the existence of a J-spectral factorization, as well as an algorithm to obtain the J-spectral factor in case it exists.     

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.     

Lower bounds for the smallest singular value

Let A = (a_ij) be an n × n complex matrix. Suppose that G(A), the undirected graph of A, has no isolated vertex. Let E be the set of edges of G(A). We prove that the smallest singular value of A, σ_n, satisfies: σ_n ≥ min σ_ij | (i, j) ∈ E, where g_ij ≡ a_i + a_j − [(a_i − a_j)² + (r_i + c_i)(r_j + c_j)]^1/2/2 with a_i ≡ |a_ii| and r_i,c_i are the i^th deleted absolute row sum and column sum of A, respectively. The result simplifies and improves that of Johnson and Szulc: σ_n ≥ min_i≠j σ_ij. (See [1].)     

On simplifying assumptions of Runge-Kutta methods for index 2 differential algebraic problems

In this paper we show a result that ensures certain order for the local error Runge-Kutta methods for index 2 differential algebraic problems with the help of the simplifying conditionsB(p),C(q),D(r) andA ₁(s) for the differential component andB(p), C(q), andA ₂(s) for the algebraic component.     

One step integration methods with maximum stability regions

One step integration methods, using K function evaluations, for the solution of a system of ordinary differential equations dv/dt=A?v are evaluated. A general expression for a class of methods is found for all positive integers K. This class of methods is proven to maximize the interval on the imaginary axis which is contained in the stability region such that the stability constraint is optimized. It is proven that every method with this optimal stability property has a polynomial M defined by y_1+,Δt=M?v_i for which M(iS_l)=exp(iS_lπ/2) where S_l=K?l.     

On the structure of generalized rough sets

In this paper we consider some fundamental properties of generalized rough sets induced by binary relations on algebras and show that

1.

Any reflexive binary relation determines a topology.

2.

If θ is a reflexive and symmetric relation on a set X, then O={A⊆X|θ_-(A)=A} is a topology such that A is open if and only if it is closed.

3.

Conversely, for every topological space (X,O) satisfying the condition that A is open if and only if it is closed, there exists a reflexive and symmetric relation R such that O={A⊆X|R_-(A)=A}.

4.

Let θ be an equivalence relation on X. For any pseudo ω-closed subset A of X, θ₋(A) is an ω-closed set if and only if ω(x, x, … , x) ∈ θ₋(A) for any x ∈ X.

Moreover we consider properties of generalized rough sets.     

An approximation method for high-order fractional derivatives of algebraically singular functions

The fractional derivative D^qf(s) (0≤s≤1) of a given function f(s) with a positive non-integer q is defined in terms of an indefinite integral. We propose a uniform approximation scheme to D^qf(s) for algebraically singular functions f(s)=s^αg(s) (α>−1) with smooth functions g(s). The present method consists of interpolating g(s) at sample points t_j in [0,1] by a finite sum of the Chebyshev polynomials. We demonstrate that for the non-negative integer m such that m<q<m+1, the use of high-order derivatives g⁽ⁱ⁾(0) and g⁽ⁱ⁾(1) (0≤i≤m) at both ends of [0,1] as well as g(t_j), t_j∈[0,1] in interpolating g(s), is essential to uniformly approximate D^q{s^αg(s)} for 0≤s≤1 when α≥q−m−1. Some numerical examples in the simplest case 1<q<2 are included.     

Direct connection between time-domain performance and frequency-domain characteristics

Presented are the relationships between the transient response characteristics of a closed-loop system due to a step input and the frequency response characteristics which influence this response. The class of transfer functions addressed include those which are RH ₂. The relationships developed are for the problem: Given: Y(s)=H(s)R(s) where H(s)∈ RH ₂ and R(s) is γs. Find:The relationship ∥y(t)∥_∞⩽Λγ∥H(jω)∥_∞ where the scaling factor Λ is a function of the frequency-domain characteristics of H(s). These relationships are useful for feedback control in that hard time-domain constraints can be enforced as amplitude and phase conditions on the open-loop transfer function. Two applications using these relationships are developed. The first application demonstrates controller design for an actuator saturation constraint. The second application is the development of a performance prediction technique for minimum phase, stable regulating systems. © 1998 John Wiley & Sons, Ltd.     

PID controller design of TITO system based on ideal decoupler

The proposed method for designing multivariable controller is based on ideal decoupler D(s) and PID controller optimization under constraints on the robustness and sensitivity to measurement noise. The high closed-loop system performance and robustness are obtained using the same controller in all loops. The method is effective despite the values and positions of the right half plane zeros and dead-times in the process transfer function matrix G_p(s). The validity of the proposed multivariable control system design and tuning method is confirmed using a test batch consisting of Two-Input Two-Output (TITO) stable, integrating and unstable processes, and one Three-Input Three-Output (TITO) stable process.     

Design of robust PID controllers using decoupling at singular frequencies

Algorithms for computation of the total set of PID stabilizers based on decoupling of PID parameter space at singular frequencies are presented. Nonconvex stability regions are built up by convex polygonal slices. Two problems are thereby discriminated: (A) assertion of k_P-intervals with stable polygons and (B) automatic detection of stable polygons for a fixed k_P. This paper includes solutions to both problems. The methods apply also for robust PID stabilization of a multi-model uncertainty.     

Classification of dynamic processes and PID controller tuning in a parameter plane

A quadruplet, defined by the ultimate frequency ω_u, the ultimate gain k_u, the angle φ of the tangent to the Nyquist curve at the ultimate frequency and the gain G_p(0), is sufficient for classification of a large class of stable processes, processes with oscillatory dynamics, integrating and unstable processes G_p(s). From the model defined by the above quadruplet, a two parameter model G_n(s_n) is obtained by the time and amplitude normalizations. Two parameters of G_n(s_n), the normalized gain ρ and the angle φ, are coordinates of the classification ρ-φ parameter plane. Model G_n(s_n) is used to obtain the desired closed-loop system performance/robustness tradeoff in the desired region of the classification plane. Tuning procedures and tuning formulae are derived guaranteeing almost the same performance/robustness tradeoff as obtained by the optimal PID controller, applied to G_p(s) classified to the same region of the classification plane. Validity of the proposed method is demonstrated on a test batch consisting of stable processes, processes with oscillatory dynamics, integrating and unstable processes, including dead-time.     

A simplified approach for the design of basically non-interacting multiple-input-multiple-output systems using qft

Quantitative feedback theory (QFT) has presented techniques for the design of multiple-input-multiple-output (MIMO) linear time invariant (LTI) systems with structured parameter uncertainty in the plant P for the satisfaction of specifications on the closed loop transfer function matrix T = [T_ij]. In many practical applications the specifications are of the basically non-interacting (BNIA) type, i.e. a_ii(ω) < | T_ii(jω) | < b_ii(ω), | T_ij(jω) | < b_ij(ω), (i ≠ j) and b_ij(ω) < a_ii(ω) in a significant range of frequencies. In one QFT technique the design is based on expressing when the matrix of compensators G = diag(G_ii(s)), L_i = G_iiQ_ii, P^?1 = [1/Q_ij], D_ij a disturbance due to plant interaction between the different system channels. It is shown in this paper that when the specifications are BNIA and F = diag(F_ii(s)), the effect of the disturbance acting on the main diagonal terms (i.e. D_ii) can be neglected. This observation saves some computational burden because satisfaction of specifications on the T_iis becomes a single-input-single-output (SISO) design problem instead of the more elaborated multiple-input-single-output (MISO) design problem which had to be designed originally. A detailed 2-input-2-output design example is presented illustrating the simpler approach, stressing the importance of considering the correlation between specifications in the design procedure.     

Rank of convex combinations of matrices

LetA andB be complex rectangular matrices of the same rankr. We characterize the property that all convex combinations ofA andB are of rankr. Moreover, forA andB of full rank, some conditions for the matrix setr(A, B) (c(A, B), resp.) whose rows (columns, resp.) are independent convex combinations of the rows (columns, resp.) ofA andB are also proposed.     

Finding Maximum Edge Bicliques in Convex Bipartite Graphs

A bipartite graph G=(A,B,E) is convex on B if there exists an ordering of the vertices of B such that for any vertex v??A, vertices adjacent to v are consecutive in?B. A complete bipartite subgraph of a graph G is called a biclique of G. Motivated by an application to analyzing DNA microarray data, we study the problem of finding maximum edge bicliques in convex bipartite graphs. Given a bipartite graph G=(A,B,E) which is convex on B, we present a new algorithm that computes a maximum edge biclique of G in O(nlog?³ nlog?log?n) time and O(n) space, where n=|A|. This improves the current O(n ²) time bound available for the problem. We also show that for two special subclasses of convex bipartite graphs, namely for biconvex graphs and bipartite permutation graphs, a maximum edge biclique can be computed in O(n??(n)) and O(n) time, respectively, where n=min?(|A|,|B|) and ??(n) is the slowly growing inverse of the Ackermann function.     

Bounded solutions of nonlinear difference equations with advanced arguments

Sufficient conditions of existence and uniqueness of α-bounded and bounded solutions to the difference equation with advancedd arguments x(n + 1) = A(n)x(n) + B(n)x(σ₁(n)) + f(n, x(n), x(σ₂(n)), σ_i(n) ⩾ n + 1, i = 1,2, are given. It is proven that under certain conditions it is possible to find positive numbers R, μ, such that from every initial condition ξ satisfying ∥ξ∥ ⩽ R, a unique bounded solution, belonging to the ball ∥x∥ ⩽ μ, starts.     

Quantitative feedback design for sampled-data systems

In quantitative feedback theory (QFT) the plant uncertainty is defined by a set P = {P} ofpossible plants. The problem is to guarantee that the system response is in a specified acceptable set A, for all P in P. QFT has been developed for large classes of plants imbedded in continuous feedback structures. This paper extends QFT to sampled-data structures. A central problem is to find the minimum sampling frequency (ω_s)_min needed. The greater the plant uncertainty and the narrower the performance tolerances, the larger must ( ω_s)_min be. The detailed design procedure parallels very closely that for continuous systems, by using the complex variable w, which maps the unit circle in the z-domain to the imaginary axis in the w-domain.     

Aligning Two Convex Figures to Minimize Area or?Perimeter

Given two compact convex sets P and Q in the plane, we consider the problem of finding a placement ? P of P that minimizes the convex hull of ? P∪Q. We study eight versions of the problem: we consider minimizing either the area or the perimeter of the convex hull; we either allow ? P and Q to intersect or we restrict their interiors to remain disjoint; and we either allow reorienting P or require its orientation to be fixed. In the case without reorientations, we achieve exact near-linear time algorithms for all versions of the problem. In the case with reorientations, we compute a (1+ε)-approximation in time?O(ε ^?1/2log?n+ε ^?3/2log? ^a (1/ε)) if the two sets are convex polygons with n vertices in total, where a∈{0,1,2} depending on the version of the problem.