Staggered sampling to improve stability of multiple-sampler feedback systems

Basing on lifting technology, this paper researches a design method for multirate sampled-data control system in the background of flight control system design, which consummates the theory of multirate sampled-date control system design for further steps, and provides a method to design multirate sampled-data controller. As the reason of uncertainty in the aircraft model and the need of different sampling frequencies in different control loops, H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> -optimal multirate sampled-data control is considered. First, multirate sampled-data control system is outlined, and then the signal description are presented. Because of multirate sampled-data system is time-variant, the direct digital controller design becomes difficult. The equivalent discrete time-invariant systems can be converted from the linear time-variant systems through lifting technology. Then the H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> discretization of multirate sampled-data systems is researched. After the sampled-data control system was lifted, the equivalent L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> [0, h) -induced norm is proven. Base on the above, the method of H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> -optimal multirate sampled-data control is summarized. A height control is simulated using a simplified aircraft model to illustrate the design. The simulation shows that the multirate Sampled-data control system can achieve the satisfactory objective accurately and optimally.

Some extensions on recent work in passive stability theory by Desoer and the author are presented. The stability of feedback systems is considered where the forward-loop and return-loop subsystems each have been subdivided into several systems operating in parallel. Results are obtained in the areas of Lyapunov stability, L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> input L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> output stability, and bounded-input bounded-state stability. Before applying these results to a specific multiloop feedback system, a pseudoenergy analysis must be done on each sub sub-system. Two specific types of systems are so analyzed. The first is a general linear first-order time-varying system. The second is a linear time-varying infinite-dimensional system; the analysis of this system takes the form of a Kalman-Yacubovich-type lemma. By using these two new systems, along with several others that have been analyzed previously, stability theorems for many specific multiloop feedback systems can be proven. One such example is given.

In this paper, we study robust stability and stabilizability of sampled-data control systems composed of a continuous-time plant and a discrete-time compensator together with sample/hold devices, using the frequency domain method. Our study is not confined only to the case of standard (i.e., single-rate) sampled-data controllers but also extended to the case of multirate sampled-data controllers. Specifically, we compare, by numerical examples, the tolerable amounts of the uncertainty of a given continuous-time plant for which the robust stabilization can be attained in the three cases: the cases of a continuous-time controller, a single-rate sampled-data controller, and a multirate sampled-data controller. The result will be useful for estimating the deterioration caused by use of sampled-data controllers in place of continuous ones, or evaluating the advantages of the multirate sampled-data scheme over the standard one.

The robust stability criterion for sampled-data control systems with a sector nonlinearity was presented in our previous papers (1996, 1998). Although it is applicable only to the sampled-data control system of a certain class, a usual discrete-time control system belongs to this type of class. The criterion contains the robust stability condition in regard to the nonlinear allowable band (sector) for continuous-time systems and the linear allowable gain band (interval set) for discrete-time systems. The paper focuses primarily on the concept of robust stability in order to analyze a nonlinear element by the linear gain band, in other words, the validity of Aizerman's conjecture extended into discrete-time systems.

This paper overviews the research investigations pertaining to stability and stabilization of control systems with time‐delays. The prime focus is the fundamental results and recent progress in theory and applications. The overview sheds light on the contemporary development on the linear matrix inequality (LMI) techniques in deriving both delay‐independent and delay‐dependent stability results for time‐delay systems. Particular emphases will be placed on issues concerned with the conservatism and the computational complexity of the results. Key technical bounding lemmas and slack variable introduction approaches will be presented. The results will be compared and connections of certain delay‐dependent stability results are also discussed.

Exponential stability for nonlinear fractional order sampled-data control systems with its applications

Sampled-data control systems generally have fixed sampling frequencies which must be set high enough to give satisfactory performance for all anticipated conditions. A study is made here of an adaptive system which varies the sampling frequency by measuring a system parameter. It is shown that a sampler followed by a zero-order hold whose sampling period is controlled by the absolute value of the first derivative of the error signal will be a more "efficient" sampler than a fixed-frequency sampler. That is, over a given time interval, fewer samples are needed with the variable-frequency system than with a fixed-frequency system while maintaining essentially the same response characteristics. Analog computer studies of simple Type I and Type II sampled-data servo systems with error sampling and unity feedback verified the method. Standard analog computer components were used to set up a simulated servo system, a rate detector, absolute-value detector, a voltage-controlled oscillator, and a sampler and zero-order hold. The system described reduced the number of samples required for response to a step input to about three-quarters that required in a fixed-sampling-frequency system. Over a long period of time savings in the number of samples required can be expected to be between 25 and 50 per cent. In many applications the savings produced by reducing the over-all number of samples required may outweigh the added complexity of the adaptive-sampling-frequency system.

Many of nonlinear control systems are sampled-data system, i.e. the continuous-time nonlinear plants are controlled by digital controllers. So it is important to investigate that if the solution of the discrete-time output regulation problem is effective to sampled-data nonlinear control systems. Recently a feedforward neural network based approach to solving the discrete regulator equations has been presented. This approach leads to an effective way to practically solve the discrete nonlinear output regulation problem. In this paper the approach is used to sampled-data nonlinear control system. The simulation of the sampled-data system shows that the control law designed by the proposed approach performs much better than the linear control law does.

The decentralized \({\mathscr{H}}_{\infty }\) sampled-data control problem is investigated for a class of continuous-time large-scale networked nonlinear systems with interconnection. Each nonlinear subsystem in the considered large-scale system is represented by a Takagi–Sugeno model and is closed by a communication channel with transformed time delay. Our objective is to design a decentralized sampled-data fuzzy controller such that the resulting fuzzy control system is asymptotically stable with an \({\mathscr{H}}_{\infty }\) performance. Firstly, using an input delay approach, the sampled-data control system is formulated into the system with time-varying delay, and a two-term approximation method is proposed such that the delayed system is reformulated into an interconnected framework with input and output. Then, we introduce a Lyapunov–Krasovskii functional that all Lyapunov matrices are no longer required to be positive definite. Combined with the scaled small gain theorem, the less conservative solutions to the decentralized \({\mathscr{H}}_{\infty }\) sampled-data control problem for the considered system are derived in the form of linear matrix inequalities. Finally, the effectiveness of the proposed methods is illustrated by two numerical examples.

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs the existence of an ISS-Lyapunov function implies the input-to-state stability of a system. Then for the case of systems described by abstract equations in Banach spaces we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system which linear approximation is ISS. In order to study interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.

The effect of weighted input and output spaces on the stability of feedback systems is examined. For a weighted space, the recent past of a signal is emphasized and the remote past is deemphasized. Tapered spaces are a subclass of weighted spaces. Results concerning weighted spaces used on linear systems are given. The circle conditions are recovered for feedback systems with exponentially tapered spaces.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The paper proposes a method for optimizing sampled-data systems over a set of reduced order controllers. The method is based on the concept of parametric transfer function and finding a subset of solutions for polynomial Diophantine equation with the order not exceeding the preset number. The method is applied to optimization of H2-norm of sampled-data submarine motion control system.

Output feedback stabilization for Markov-based nonuniformly sampled-data networked control systems

A novel linear matrix inequality (LMI) condition for the stability of the sampled-data fuzzy control system based on the Takagi-Sugeno fuzzy model is presented. Using the novel Lyapunov functional, the relaxed stability condition is presented for the sampled-data fuzzy control and represented in the LMI format. A simulation example is provided to verify the effectiveness of the proposed technique.

An improved digital redesign for sampled-data fuzzy control systems: Fuzzy Lyapunov function approach