An algebraic algorithm for determining the desired gain of multivariable feedback systems

It is shown that the return difference matrix possesses the same functions and plays the same important role in stability analysis and design of a multivariable feedback network as it does in control theory. The generalized Nyquist criterion, characteristic loci and other methods of stability analysis and design for multivariable feedback systems in control theory are applicable directly to multivariable feedback networks. In particular, a general frequency stability criterion for a multivariable lumped and distributed-parameter feedback network is presented.

In this chapter, techniques are described for assessing the relative stability of a multivariable control system design. An interesting and important feature in multivariable feedback systems is that they have different stability margins at different points in their configuration. According to the generalized Nyquist stability criterion the feedback system is stable if, and only if, the number of anticlockwise encirclements of the critical point -1 by the characteristic gain loci, is equal to the number of open-loop unstable poles.

The purpose of this short paper is to discuss generalizations of the concept of phase from single loop to multiple loop feedback systems.

This paper examines the limitations imposed by Right Half Plane (Rhp) zeros and poles in multivariable feedback systems. The main result is to provide lower bounds on || WXV(s) || <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> where X is S, S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</inf> , T or T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</inf> (sensitivity and complementary sensitivity). Previously derived lower bounds on the H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> -norm of S and T are thus generalized to the case with matrix-valued weights, including bounds for reference tracking and disturbance rejection. Furthermore, new bounds which quantify the minimum input usage for stabilization in the presence of measurement noise and disturbances are derived.

Methods for the design and tuning of multivariable model state feedback (MMSF) systems for complete and triangular output decoupling of inherently stable processes are presented. MMSF is a method of implementing multivariable IMC controllers that substantially reduces the complexity of the implementation and compensates for control effort saturation. Complexity is reduced because the controls are formed as a linear combination of past and current model states. Saturation‐induced directionality problems are avoided by temporarily increasing filter time constants to bring the control vector into the constraint set. The possibility that needed future controls will be inadequate to compensate for past control actions is avoided by setting minimum values for the controller filter time constants. The feedback structure of MMSF facilitates on‐line tuning, one process variable at a time. Also, because MMSF is a form of IMC, it can be tuned off‐line to accommodate process uncertainty using H ∞ methods.

In this chapter, the concepts described in Chaps. 2 and 3 are extended to multivariable and multi-loop feedback systems. For the input-output stability of multivariable systems, matrix representations and inequalities based on the norms in the l 2 space will be defined. Usually, multivariable control systems are analyzed in a state space and are designed by using quadratic forms. However, a design method on the basis of vector-matrix representations, e.g., the linear-quadratic (LQ) form, is not practical, because controlled systems are accompanied by nonlinearities. Therefore, also in this chapter, for multiple nonlinearities, inequality conditions and general expressions for the M-matrix are applied to the stability problem for multi-loop control systems.

Subsystem identification of multivariable feedback and feedforward systems

Statistical bounds for model parameters and corresponding system frequency-response limits are established. The results are extended to multivariable systems to produce an element-by-element characterisation of system uncertainty. This uncertainty representation can be used to develop frequency-response bounds on the eigenfunctions of perturbed multivariable systems; thus producing system gain and phase information via a generalised Nyquist analysis. An application of the technique is presented using a two-input/two-output system.

The root-locus-approximation procedure is extended to provide for the complex-plane analysis of equations which contain a term quadratic in the variable parameter in addition to the usual linear term. Such an extension will have many applications; in the paper, a discussion is given concerning the design of linear time-invariant multivariable feedback systems. It is shown that, particularly for 2-input/2-output problems, where the algebra involved in obtaining the equations to be analysed remains fairly simple, much useful information is available by straightforward hand calculation. Brief suggestions are given for further generalisations of the techniques (inclusion of a cubic gain term etc.)

The Computer Aided Design of Multivariable Non Linear Control Systems using Frequency Domain Techniques

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

The μ analysis method is applied to a class of multivariable separable autonomous nonlinear feedback systems. The nonlinear element is replaced by its sinusoidal input describing function (SIDF) to develop a quasilinear nominal model for control analysis and design. Some properties of the ∞ -norm and structured singular values (SSVs) are shown to hold for the type of systems where the output is a function of control input-signal amplitude and frequency. The model uncertainty induced by the SDIF approximation is calculated and incorporated in the control design as an input multiplicative perturbation to the quasilinear nominal model. The μ analysis technique is then employed to study the stability and performance robustness of the nonlinear control system. Using small gain theorem, the condition for the absence of limit cycle is derived. The information obtained from this analysis is then used to design a bank of controllers using the H ∞ optimization-based μ synthesis technique. These controllers are designed such that the closed loop system is robust over a range of control-input amplitude. A simple scheme is then proposed to select the appropriate controller for the operating range. The simulation results illustrate that the proposed technique may be used to design a high performance robust controller for nonlinear systems and the absence of the limit cycle oscillation may be ensured.

Revisiting the stability of the HIE-FDTD technique for modeling graphene dispersion

In this paper, a new example of applying Routh-Hurwitz stability criterion is provided to design an optimal PD control gains for stabilizing a TORA. We solve analytically an optimization problem, which achieves a fast control response of the TORA. We present two steps to solve the optimization problem analytically. First, we perform a coordinate transformation to the TORA's characteristic equation, and prove its unstability. Second, we design the optimal control gains which assign the unstable poles to the imaginary-axis by using the Routh-Hurwitz criterion. By using these two steps, we provide the optimal control gains and show that the closed-loop system consisting of the TORA and the PD controller has two pairs of complex-conjugate poles with the same real part. We verify the validity of the analytical result by comparing with numerical computation and numerical simulation.

A multivariable feedback system y¯(s) = G(s)x(s). x¯(s)=ū(s) − F(s)y(s) is considered, where G(s δ (gkl(s) is the n × n plant transfer matrix and f(s) = diag (f 1) [tdot] f 2(s) [tdot] f n(s)). By imposing some restrictions on the structure of G(s), we may use the eigenvalues λj(s), 1≤j≤n of G(s) to infer the stability of the closed-loop system in an arbitrary rectangular region Φ δ{F − ∞αk≤ƒkβk≤∞, 1≤k≤n}, which is a set of positive Lebesgue measure. The stability result contains Rosenbrock's stability theorem as a special case.