Time-invariant Feedback Systems Research Articles

Addendum to “Converse Theorems for Integral Quadratic Constraints”

Consider a linear time-invariant feedback system consisting of two open-loop stable subsystems. In this note, it is established that if such feedback system is stable for one subsystem being any arbitrarily passive system, then the other subsystem must be output strictly passive.

Controller Design for Diagonal Decoupling and Integral Action

In the standard linear, time-invariant feedback system, controllers that achieve diagonal decoupling and closed-loop stability exist if and only if the plant satisfies the diagonal denominator condition or has no coinciding poles and zeros in the region of instability. A simple and systematic decoupling design procedure is presented under each of these conditions. The closed-loop poles can be placed at any desired points, and free parameters are included for satisfying additional design objectives. The designs are also extended to provide integral action in order to track step inputs with zero steady-state error.

A New Anti-Windup Compensator Based on Quantitative Feedback Theory for an Uncertain Linear System with Input Saturation

This paper devotes to the robust stability problem for an uncertain linear time invariant (LTI) feedback system with actuator saturation nonlinearity. Based on a three degree of freedom (DOF) non-interfering control structure, the robust stability is enforced with the describing function (DF) approach for an uncertain LTI system to avoid the limit cycle. A new type of anti-windup (AW) compensator is designed using the quantitative feedback theory (QFT) graphical method, which results in a simple design procedure and low-order AW control system. One of the most significant benefits of the proposed method is free of the non-convexity (intractable) drawback of the linear matrix inequality (LMI)-based approach. The analysis conducted on the benchmark problem clearly reveals that the proposed QFT-based anti-windup design is able to handle both saturation and uncertainty in a very effective manner.

Mean-square stabilizability via output feedback for a non-minimum phase networked feedback system

This work studies mean-square stabilizability via output feedback for a networked linear time invariant (LTI) feedback system with a non-minimum phase plant. In the feedback system, the control signals are transmitted to the plant over a set of parallel communication channels with possible packet dropout. Our goal is to analytically describe intrinsic constraints among channel packet dropout probabilities and the plant’s characteristics in the mean-square stabilizability of the system. It turns out that this is a very hard problem. Here, we focus on the case in which the plant has relative degree one and each non-minimum zero of the plant is only associated with one of control input channels. Then, the admissible region of packet dropout probabilities in the mean-square stabilizability of the system is obtained. Moreover, a set of hyper-rectangles in this region is presented in terms of the plant’s non-minimum phase zeros, unstable poles and Wonham decomposition forms which is related to the structure of controllable subspace of the plant. A numerical example is presented to illustrate the fundamental constraints in the mean-square stabilizability of the networked system.

Identification of heart rate dynamics during moderate-to-vigorous treadmill exercise.

BackgroundHeart rate can be used to prescribe exercise intensity for development and maintenance of cardiorespiratory fitness. The aim of this study was to identify the dynamics of heart rate response during moderate-to-vigorous treadmill exercise and to explore parameter dependencies with respect to time, intensity level and step-change direction. The focus was on simple approximate models for subsequent design of heart rate control systems.Methods24 healthy, able-bodied male subjects each did two separate, 35-min tests on a treadmill, one at moderate and one at vigorous intensity. Each test had four individual upward and downward steps (1–4). Heart rate responses were modelled as first-order transfer functions with steady-state gain k and time constant tau. Models were estimated both for the overall testing periods and for individual step responses within each test.ResultsThere were no significant differences in the overall mean values of k [24.3 vs. 24.1 bpm/(m/s), p = 0.88] and tau (55.7 vs. 59.5 s, p = 0.53) between the two intensity levels. The overall nominal gain for both conditions was k = 24.2 pm 8.3, 21.9–26.6 bpm/(m/s) (mean pm standard deviation, 95 % confidence interval), and the overall nominal time constant was tau = 57.6 pm 23.6, 50.9–64.3 s. Analysis of models estimated from the individual steps revealed a significant difference in steady-state gain k for upward and downward steps [30.2 vs. 23.6 bpm/(m/s), p < 0.001], but no difference in time constant tau between these two directions (57.5 vs. 54.4 s, p = 0.52). For gain k, there was no significant main effect of intensity (p = 0.35) or intensity–time (p = 0.86) interactions, but there was a significant main effect of time (p < 0.001). Pairwise comparison with respect to time showed a significant difference between the upward steps at times 1 and 3 [33.0 vs. 27.3 bpm/(m/s), p < 0.01], but no significant difference between the downward steps at times 2 and 4 [24.4 vs. 22.8 bpm/(m/s), p = 1.0]. For time constant tau, there were no significant main effects of intensity (p = 0.36) or time (p = 0.89), or intensity–time interactions (p = 0.23).ConclusionsThe tight CI-bounds obtained, and the observed parameter dependencies, suggest that the overall nominal model with k = 24.2 and tau = 57.6 might serve as the basis for design of a linear time-invariant (LTI) feedback system for real-time control of heart rate. Future work should focus on this hypothesis and on direct comparison of LTI and nonlinear/time-varying control approaches.

Generalizing Nyquist criteria via conformal contours for internal stability analysis

By contriving the regularized return difference relationship in linear time-invariant (LTI) feedback systems, we attempt to generalize and validate the Nyquist approach for such internal stability as Lyapunov stability/instability, asymptotic stability, exponential stability and district stability (or -stability), respectively, even when there exist decoupling zeros, by means of what we call the regularized Nyquist loci that are plotted with respect to a Nyquist contour and its conformal one(s). More precisely, miscellaneous open-loop/closed-loop pole cancellations in the return difference relationship that may complicatedly tangle our stability interpretation but usually neglected in most existing Nyquist criteria are scrutinized. And then, Nyquist-like criteria for internal stability are claimed with the regularized Nyquist loci. These criteria get rid of pole cancellations testing and can be implemented completely independent of open-loop pole distribution knowledge; moreover, the Nyquist criteria for asymptotic/exponential stability are necessary and sufficient, while those for -stability are sufficient. Internal stability of a cart system with an inverted pendulum is examined to illustrate the results.

Frequency-domain L 2-stability conditions for switched linear and nonlinear SISO systems

We consider the L 2-stability analysis of single-input–single-output (SISO) systems with periodic and nonperiodic switching gains and described by integral equations that can be specialised to the form of standard differential equations. For the latter, stability literature is mostly based on the application of quadratic forms as Lyapunov-function candidates which lead, in general, to conservative results. Exceptions are some recent results, especially for second-order linear differential equations, obtained by trajectory control or optimisation to arrive at the worst-case switching sequence of the gain. In contrast, we employ a non-Lyapunov framework to derive L 2-stability conditions for a class of (linear and) nonlinear SISO systems in integral form, with monotone, odd-monotone and relaxed monotone nonlinearities, and, in each case, with periodic or nonperiodic switching gains. The derived frequency-domain results are reminiscent of (i) the Nyquist criterion for linear time-invariant feedback systems and (ii) the Popov-criterion for time-invariant nonlinear feedback systems with the Lur'e-type nonlinearity. Although overlapping with some recent results of the literature for periodic gains, they have been derived independently in essentially the Popov framework, are different for certain classes of nonlinearities and address some of the questions left open, with respect to, for instance, the synthesis of the multipliers and numerical interpretation of the results. Apart from the novelty of the results as applied to the dwell-time problem, they reveal an interesting phenomenon of the switched systems: fast switching can lead to stability, thereby providing an alternative framework for vibrational stability analysis.

An Average Performance Limit of MIMO Systems in Tracking Multi-Sinusoids With Partial Signal Information

This paper studies the best achievable reference tracking performance of MIMO linear time-invariant (LTI) feedback systems with partial reference information. The reference signal to be tracked is a multi-tone sinusoidal signal. It is assumed that, other than the instantaneous values of the reference signal, only the frequencies of the sinusoidal components are known. The tracking performance is measured by the energy of the tracking error. With this partial information of the reference signal, we consider an averaged performance measure and obtain an explicit expression of the best achievable performance. The expression shows how the harmonic frequencies and the zero directions may affect the performance, and further, how a performance degradation may result under the available partial information.

Loop-gain optimization of unstable-non-minimum phase feedback systems

The paper presents a design method for optimizing the loop-gain of unstable-non-minimum phase feedback systems. Because obvious gain-bandwidth limitations exist in such systems, the proposed best loop gain optimization goal is maximization of the stability gain and phase margins, which are inherently low in such feedback systems. The design procedure is based on coprime factorization of the plant and controller, and the Bezout identity. An optimization process varies iteratively the observer-based controller parameters until maximization of the gain and phase margins is achieved. The same design technique also allows one to solve the mixed sensitivity H ∞ standard control problem for linear time invariant feedback systems.

Robust control design for SISO systems based on constrained optimization

An analytical method is described for the design of linear time-invariant SISO feedback systems to meet specified performance criteria in the presence of structured and unstructured uncertainty. The design framework also provides for control over the degree of the compensator.

Inherent design limitations for linear sampled-data feedback systems

There is a well-developed theory describing inherent design limitations for linear time invariant feedback systems consisting of an analogue plant and analogue controller. This theory describes limitations on achievable performance present when the plant has non-minimum phase zeros, unstable poles, and/or time delays. The parallel theory for linear time invariant discrete time systems is less interesting because it describes system behaviour only at sampling instants. This paper develops a theory of design limitations for sampled-data feedback systems wherein the response of the analogue system output is considered. This is done using the fact that the steady-state response of a hybrid feedback system to a sinusoidal input consists of a fundamental component at the frequency of the input together with infinitely many harmonics at frequencies spaced integer multiples of the sampling frequency away from the fundamental. This fact allows fundamental sensitivity and complementary sensitivity functions that re...

The sensor noise problem in dithered feedback systems

This paper studies the problem of sensor noise in dithered adaptive systems. It has been previously demonstrated that dithered adaptive systems may have superior sensitivity reduction properties than linear time invariant (LTI) feedback systems. However it is shown in this paper that the dithered designs which contain a saturating nonlinear element may have sensor noise properties worse than the LTI design for the same uncertainty or disturbance problem, despite the considerably smaller bandwidth of the nonlinear dithered design. Two new versions of dithered adaptive systems are developed, where the nonlinear element in the main loop is eliminated, and noise is considerably reduced. Bandwidth and sensor noise effect improvement are now both significantly less than in the LTI design. Systematic design procedures are presented for satisfying quantitative system specifications for a large class of feedback problems. The resulting designs are optimum with respect to loop bandwidth and close to optimum for sensor noise effect. A numerical example not solvable with LTI design techniques is shown illustrating the design technique.

Comparison and extensions of control methods for narrow-band disturbance rejection

Techniques for designing and implementing algorithms for the control of periodic narrowband disturbances are discussed, and the strong similarities between the different methodologies are shown. The analysis of linear time invariant feedback systems results in a suggestion of how to extend two of the LQ-based multiple-input, multiple-output methodologies to achieve improved performance and stability robustness properties. In the analysis of the adaptive feedforward methods, it is concluded that the popular filtered-x LMS algorithm is useful for implementation, but is best analyzed from a classical linear time invariant feedback perspective. This perspective results in a suggestion of how to extend the multiple error LMS algorithm to achieve improved performance and stability robustness. Stability bounds of the adaptive feedforward approaches are derived in terms of allowable model error.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Robust design of MIMO feedback systems having an uncertain non-linear plant

An n × n non-linear multiple-input multiple-output system W is presented, known only to belong to a set {W}. W is embedded in a two-degree-of-freedom feedback structure, {r} is a set of possible command inputs, where for each r ∊ {r} there is a set {A r } of acceptable outputs. The two degrees of freedom have to be selected such that in response to each command input r, the plant's output y ∊ {Ar } for all W ∊ {W}, A design technique is presented for solving this problem, based on the Horowitz design method for MIMO linear time invariant feedback systems. Sufficient conditions are given under which a solution is guaranteed. A design example is provided.

Quantitative design method for MIMO uncertain plants to achieve prescribed diagonal dominant closed-loop minimum-phase tolerances

A quantitative design method for multi-input multi-output linear time-invariant feedback systems for plants with large uncertainty has been presented by Horowitz ( 1982), and by Yaniv and Horowitz ( 1986). This design method is developed here to guarantee minimum-phase closed-loop diagonal elements for systems with basically non-interacting (Horowitz and Loecher 1981) off-diagonal closed-loop tolerances. The advantage of this design is that with minimum-phase transfer functions, a very important class of time-domain specifications can be translated to the frequency domain, as shown by Krishman and Cruickshanks ( 1977) and by Horowitz ( 1976). The attractive properties of this design method are: (a) the problem is reduced to a successive single-loop design with no interaction between the loops, and no iterations are necessary; (b) the technique can be applied to all n × n plants P with P−1 having no poles in the right-half plane, and satisfying some conditions described in § 5; (c) the procedure is interac...

Algebraic theory of linear time-invariant feedback systems with two-input two-output plant and compensator

This paper present an algebraic theory for linear time-invariant multi-input multi-output systems with two-input two-output plant and compensator, using the factorization approach. This system configuration considered by Doyle and Nett is most general in that any interconnection of two systems can be represented in terms of this scheme. Other system configurations encountered in compensator design problems are special cases of this system configuration. The analysis and synthesis applies to continuous-time as well as discrete-time systems, and to lumped as well as distributed systems, since the algebraic setting is completely general. The compensator parametrization has four degrees of freedom. The paper is self-contained and is tutorial in the sense that it develops the main results of Nett without introducing the concept of containment.

Matrix factorization method to stabilize multivariable control systems

A new method to study the stability of n-input- n-output linear time-invariant feedback systems is proposed. First, a new factorization of the transfer-functions matrix is developed which allows a direct analysis of the stability of a multivariable system. Then, some properties of this factorization are studied and finally, a design method is inferred from these properties. The method permits an easy design of stable controllers. It allows the influence of the off-diagonal elements of the controller on the stability of the closed loop system to be studied.

The singular-G method for unstable non-minimum-phase plants†

Abstract In linear time-invariant feedback systems with plants which have both poles and zeros in the right half-plane, it is always possible to stabilize the system for a fixed plant. But in the previous optimum techniques, the stability margins might be so small as to render the design wholly impractical. This problem was overcome in the X-29 aircraft in a multiple-input-multiple-output (MIMO) setting, by use of a singular-G (compensation) matrix inside the loop. Excellent stability margins were then achievable over a wide plant parameter range, by means of a fixed-G compensation matrix. This paper extends the singular-G technique to the single-input-single-output (SISO) plant. The latter is converted into an equivalent N × N MIMO plant by means of N parallel independent time-varying modulators acting on the plant output, a technique previously used for non-linear network synthesis. The singular-G method is then applicable to the equivalent N × N MIMO plant. The detailed design procedure is presented by means of an example with N = 2.

The robustness optimization of a multivariable feedback system in Hankel norm space

This paper is concerned with the optimization of robustness in a multi-input multi-output linear time-invariant finite dimensional feedback system. The objective is to derive a closed-form solution to the problems of the synthesis of optimal controllers that either minimize the weighted sensitivity function or maximize the excess robust ness with respect to Hankel norm criterion. An all-pass solution for the minimum Hankel norm is employed to treat our problems. Finally, the infinite norm case is also discussed.

The problem of guaranteeing robust disturbance rejection in linear multivariable feedback systems

Abstract This paper considers a linear time invariant lumped multivariable feedback system consisting of a plant, which is subjected to perturbations, a precompensator and a feedback compensator. Given that, the nominal feedback system is exponentially stable and achieves considerable disturbance rejection, this paper establishes necessary and sufficient conditions on the nominal feedback system so that, over the prescribed class of additive/multiplicative perturbations, all perturbed feedback systems preserve stability and considerable disturbance rejection. The generalization to distributed systems and the extension to discrete systems are indicated.