Delay Independent Truncated Predictor Feedback for Discrete-Time Linear Systems

In this brief, we investigate the control of linear systems with input delay with applications to active magnetic bearings (AMBs). This study is motivated by emerging energy explorations that require compressors and other rotating machines to operate in remote locations. The separation between the machine and the control site in these applications, which may extend for several kilometers, introduces a time delay in the transmission of the control signals. Here, we study the effect of input delay in the control of AMBs in remote rotating machines and we present a truncated predictor feedback (TPF) control design to maximize the input delay that the stable closed-loop system can tolerate. The controller corresponding to the maximum input delay is obtained iteratively from the solutions to local linear matrix inequality problems. The capabilities of the designed TPF controller are first verified through simulation and then experimentally validated on an AMB test rig.

In this paper, we propose a delay independent control scheme that regulates to zero the state and the control input of a linear input delayed system whose open loop poles are at the origin or in the open left-half plane. Two main features of our control scheme are its nondistributed nature in the sense that only the current state is used in the feedback and its delay independence in the sense that no knowledge of the delay, neither its exact value nor its upper bound, is required. The main ingredients of our control scheme and the regulation proof include a design of the delay independent truncated predictor feedback law with a time-varying feedback parameter, Lyapunov function based adaptation of the time-varying parameter, a mechanism for switching between two update laws of the time-varying parameter, and the partial differential equation based analysis for delayed systems.

In this chapter, we propose a control scheme that, in the absence of any knowledge of the delay, regulates to zero the state and the control input of a linear input delayed system whose open loop poles are at the origin or in the open left-half plane. Two main features of our control scheme are its non-distributed nature in the sense that only the current state is used in the feedback and its delay independence in the sense that no knowledge of the delay, neither its exact value nor its upper bound, is required. The main ingredients of our control scheme and the regulation proof include the design of a delay independent truncated predictor feedback law with a time-varying feedback parameter, Lyapunov function based adaptation of the time-varying parameter, a mechanism for switching between two update laws of the time-varying parameter, and the partial differential equation based analysis of the closed-loop system.

This paper is concerned with observer based output feedback control of linear systems with both multiple inputs and multiple outputs delays. Our recently developed truncated predictor feedback (TPF) approach for state feedback stabilization of time-delay systems is generalized to the design of observer based output feedback controllers. By assuming that the open-loop time-delay systems are not exponentially unstable, infinite dimensional observer based output feedback controllers are constructed by using the TPF approach. Moreover, we show that the separation principle holds for the proposed observer based output feedback controllers.

A fundamental matrix based approach to design predictor-based control laws for discrete-time systems with state and input delays

To estimate the growth rate of matrix products $A_{n}··· A_{1}$ with factors from some set of matrices $\mathscr{A}$, such numeric quantities as the joint spectral radius $ρ(\mathscr{A})$ and the lower spectral radius $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})$ are traditionally used. The first of these quantities characterizes the maximum growth rate of the norms of the corresponding products, while the second one characterizes the minimal growth rate. In the theory of discrete-time linear switching systems, the inequality $ρ(\mathscr{A})<1$ serves as a criterion for the stability of a system, and the inequality $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})<1 $ as a criterion for stabilizability.Given a set $\mathscr{A}$ of $N×M$ matrices and a set $\mathscr{B}$ of $M×N$ matrices. Then, for matrix products $A_{n}B_{n}··· A_{1}B_{1}$ with factors $A_{i}∈\mathscr{A}$ and $B_{i}∈\mathscr{B}$, we introduce the quantities $μ(\mathscr{A},\mathscr{B})$ and $η(\mathscr{A},\mathscr{B})$, called the lower and upper minimax joint spectral radius of the pair $\{\mathscr{A},\mathscr{B}\}$, respectively, which characterize the maximum growth rate of the matrix products $A_{n}B_{n}··· A_{1}B_{1}$ over all sets of matrices $A_{i}∈\mathscr{A}$ and the minimal growth rate over all sets of matrices $B_{i}∈\mathscr{B}$. In this sense, the minimax joint spectral radii can be considered as generalizations of both the joint and lower spectral radii. As an application of the minimax joint spectral radii, it is shown how these quantities can be used to analyze the stabilizability of discrete-time linear switching control systems in the presence of uncontrolled external disturbances of the plant.

This brief addresses an output feedback control design for Lipschitz nonlinear systems in the presence of input delay. A nonlinear observer is introduced to estimate the system states. A truncated predictor state feedback, which is the conventional predictor state feedback law with the distributed term dropped, is implemented with the estimated states. Lyapunov-Krasovskii functionals are constructed to establish conditions under which this observer-based truncated predictor feedback law asymptotically stabilize the system at the origin. The effectiveness of the proposed method is demonstrated on the Chua's circuit.

The feedback control system is defined as the sampling of an output signal and feeding it back to the input, resulting in an error signal that drives the overall system. This dissertation focuses on the stabilization and performance of state feedback control systems. Chapters 3 and 4 focus on the feedback control protocol approaching in the multi-agents system. In particular, the global regulation of distributed optimization problems has been considered. Firstly, we propose a distributed optimization algorithm based on the proportional-integral control strategy and the exponential convergence rate has been delivered. Moreover, a decentralized mechanism has been equipped to the proposed optimization algorithm, which enables an arbitrarily chosen agent in the system can compute the value of the optimal solution by only using the successive local states. After this, we consider the cost function follows the restricted secant inequality. A dynamic event-triggered mechanism design has been proposed. By ensuring the global regulation of the distributed proportional-integral optimization algorithm, the dynamic event-triggered mechanism efficiently reduces the communication frequency among agents. Chapter 5 focuses on the feedback control protocol approaching the single-agent system. Specifically, we investigate the truncated predictor feedback control of the regulation of linear input-delayed systems. For the purpose of improving the closed-loop performance, we propose a design of the truncated predictor feedback method with time-varying feedback parameters and give the potential range of choosing the time-varying feedback parameters to replace the traditional constant low gain parameters.

Predictor-based stabilisation results are provided for nonlinear systems with input delays and a compact absorbing set. The control scheme consists of an inter-sample predictor, a global observer, an approximate predictor, and a nominal controller for the delay-free case. The control scheme is applicable even to the case where the measurement is sampled and possibly delayed. The input and measurement delays can be arbitrarily large but both of them must be constant and accurately known. The closed-loop system is shown to have the properties of global asymptotic stability and exponential convergence in the disturbance-free case, robustness with respect to perturbations of the sampling schedule, and robustness with respect to measurement errors. In contrast to existing predictor feedback laws, the proposed control scheme utilises an approximate predictor of a dynamic type that is expressed by a system described by integral delay equations. Additional results are provided for systems that can be transformed to systems with a compact absorbing set by means of a preliminary predictor feedback.

This paper studies the consensus problem for discrete-time multi-agent systems with state, input and different communication delays, where the input delays can be arbitrarily large yet exactly known. Moreover, the communication delays are different for different agents and can also be arbitrarily large. A nested predictor based protocol is established to solve the problem. It is shown that both the input and communication delays can be compensated completely. Furthermore, a linear matrix inequalities based approach is provided to design the state feedback gains. Finally, a numerical example is worked out to illustrate the effectiveness of the proposed approach.

The stabilization of dynamical systems with input delay using a generalized predictor feedback controller is investigated. With transforming the design procedure into the stabilization problem of a system controlled by a dynamic output feedback controller, predictor feedback gains are synthesized via linear matrix inequalities which are more flexible and extendable in comparison with the conventional predictor feedback method. Simulation examples demonstrate the effectiveness of the design method.

In this paper we analyse the stability robustness of linear continuous-time and discrete-time systems with structured time-varying uncertainties. The system to be controlled is described by the usual state-space formulation but includes time-varying uncertainties. The first new robust stability sufficient condition we have derived in this paper may become the necessary and sufficient condition of system stability in some situations. The second new condition is always less conservative than the sufficient condition recently proposed by Sobel et al., (1989). The third new condition, using the similarity transformation, provides the possibility of getting even less conservative results. In deriving these new conditions we also present some new lemmas which are valuable in related research. The results are extended to linear discrete-time systems and some illustrative examples are given.

This communication is concerned with stabilization via output static feedback of discrete-time linear time-invariant finite-dimensional systems. A new concept is introduced: output static stabilization in the relaxed sense. This (conservative) stabilization concept is defined so that it is characterized in terms of the discrete-time LQR problem. Properties, and alternative equivalent definitions of this new concept are analyzed. It is shown that for a particular class of plants, the stabilization problem in the above new sense can be cast as a convex programming problem (which moreover can be used as a tool for synthesis). A full characterization for the class of plants that are stabilizable in the above sense is presented. Important consequences of this characterization are somehow surprising. It is shown that the introduced new (conservative) stabilization concept is indeed much more important than it seems. The general problem of output static stabilization can be transformed into the stabilization problem in this new sense. A relationship between output static stabilizability and the discrete-time LQR problem is proved. Convex necessary and sufficient conditions, for joint stabilization (of multiple plants) in the above sense and for decentralized stabilization in the above sense, are presented.

The question of power and asymptotic stability of infinite dimensional discrete-time state space systems is investigated. It is shown that every balanced realization is asymptotically stable. Conditions are given for balanced, input normal, or output normal realization to be asymptotically and/or power stable. >

The paper presents the algorithm for robustness estimation of large deviations in free motion of linear discrete-time systems. A tracking discrete-time systems with the modal control law are discussed, where the modal control law is designed taking into account the value of delay and the deviation. Two cases of discrete-time systems with control signal delay are considered in the paper. The first case relates to predictable delay of control signal, and the second one relates to unpredictable delay of control signal. It is assumed that parameters of the system are linearly dependent on the uncertainties. The problem is solved with the state space approach and the sensitivity theory methods. An upper bound estimation of trajectory deviations for discrete-time systems is obtained. The estimation contains the condition number of the eigenvectors matrix of the system state matrix. Therefore, sensitivity functions of singular values of the eigenvectors matrix are used to calculate the robustness estimation of the deviations. Based on the obtained equations, an algorithm for the robustness estimation of large deviations in linear discrete-time systems with parametric uncertainties is proposed. The proposed method is compared with an alternative LMI based approach for the peak estimation problem. Advantages and disadvantages of both approaches are illustrated with examples.KeywordsDiscrete-time systemDelayLarge deviationsPeak effectSensitivity analysisUpper boundsRobustnessParametric uncertainty