Decentralized constrained control of delayed discrete-time systems

This paper addresses controller design for discrete-time positive systems with multiple delays. The control is under positivity boundedness constraints, which means that the resulting closed-loop systems are not only stable, but also positive and bounded by a given boundary, if the initial condition is bounded. The contribution lies in two aspects. First, two necessary and sufficient conditions are established, determining whether the trajectory of a positive system can be bounded by a given boundary, provided that the initial condition is bounded. Second, two necessary and sufficient condition are provided for bounded control. All the controllers are explicitly constructed if existent, and all the results are formulated as linear programming or linear matrix inequality problems, hence easy to be verified.

In this note, we deal with the stability and stabilization problems for nonlinear quadratic discrete-time periodic systems. By using the quadratic Lyapunov function, and a so called periodic invariant set, delay-independent sufficient conditions for local stability and local stabilization for nonlinear quadratic discrete-time periodic systems are derived in terms of linear matrix inequalities (LMIs). Based on these sufficient conditions, iterative linear matrix inequality algorithms are proposed for maximizing the stability regions of the systems. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.

The global stability of positive discrete-time time-varying nonlinear systems with time-varying scalar feedbacks is investigated. Sufficient conditions for the asymptotic stability of discrete-time positive time-varying linear systems are given. The new conditions are applied to discrete-time positive time-varying nonlinear systems with time-varying feedbacks. Sufficient conditions are established for the global stability of the discrete-time positive time-varying nonlinear systems with feedbacks.

This paper considers the problem of constrained stabilization of linear continuous-time systems when a quadratic cost criterion is imposed in the design of the state feedback control law. The linear quadratic regulator (LQR) is solved under positivity constraint, which means that the resulting closed-loop systems are not only optimally stable, but also positive. We focus on the class of linear continuous-time positive systems (Metzlerian systems) and use the interesting properties of Metzler matrices to provide the necessary ingredients for the main results of the paper. A two-step procedure is proposed to solve the problem. First, some necessary and sufficient conditions are presented for the existence of controllers satisfying the Metzlerian constraint, and the constrained stabilization is solved using linear programming (LP) or linear matrix inequality (LMI). Second, a sufficient condition is outlined for the existence of a solution to maintain the positivity of the first step while achieving the optimality of LQR. Finally, the robustness of the design is analyzed and possible extension of the design is proposed for an uncertain interval systems. A numerical example is included to illustrate the procedure.

2021 Benjamin Franklin Medal in Mechanical Engineering presented to C. Daniel Mote, Jr., Ph.D.

The main aim of this paper is to study the stability problem for neutral-type Hopfield neural networks possessing multiple time delays in the states of the neurons and multiple neutral delays in time derivative of states of the neurons. By using a suitable Lyapunov functional, a novel sufficient stability condition is obtained for global asymptotic stability of neutral-type neural networks with multiple delays. The derived stability criterion can be expressed in terms of the parameters of the neural network model which totally relies on some simple relationships established between the network parameters and it is completely independent of time delays and neutral delays. Hence, this new global asymptotic stability condition can be easiliy tested and verified by using some algebraic mathematical properties. We will also make a comparision between the result of this paper and previously published corresponding results. This comparision will indicate the advanteges of our proposed stability condition over the previously reported stability conditions. Since obtaining stability conditions for neutral type neural networks with multiple delays is a difficult task to achieve due to the insufficient mathematical methods and techniques, the result given in this paper can be considered an important and alternative result for this class of neutral type neural systems.

The robust stability analysis for discrete large-scale uncertain systems with multiple time delays is addressed in this paper. We establish a method for selecting properly a positive definite matrix Q to derive a very simple upper solution bound of the discrete algebraic Lyapunov equation (DALE). Then, using the Lyapunov equation approach method with this upper bound, several sufficient conditions are presented to guarantee the robust stability of the overall systems. Comparisons between the proposed results with a previous one are also given.

This paper studies the exponential stability of a class of discrete-time piecewise-linear systems (DPLS). Some basic properties of the proposed DPLS are established, which enables the generating function approach to be used for the system stability analysis. By introducing the generating functions of DPLS and showing their properties, a sufficient and necessary condition for the exponential stability of DPLS is derived. Furthermore, the maximum exponential growth rate of system trajectories can be obtained exactly by computing the radii of convergence of the generating functions. The algorithm for computing the generating functions is developed and two examples are given to demonstrate the proposed approach.

A sufficient condition for the kinematic stability of a robot hybrid position/force control scheme is defined. The test for the sufficient condition is a technique that uses only the kinematic information to determine system stability or to identify situations where possible stability problems may occur. The sufficient condition test is applied to an example, using the original hybrid position/force control scheme of J.J. Craig and M.H. Raibert (1979) and the corrected hybrid position/force control scheme proposed by W.D. Fisher and M.S. Mujtaba (1992). A root locus analysis showed the original hybrid control approach to be unstable. It is shown that the sufficient condition is also violated for this situation. In the corrected hybrid control approach, the sufficient condition is never violated for any manipulator and the same example is shown to be always stable. A root locus plot for the hybrid control approaches is included to confirm the results of the sufficient stability condition. >

Multiple summation inequalities and their application to stability analysis of discrete-time delay systems

This paper is concerned with the control of systems composed of multiple coupled subsystems. In such architectures, communication between different local controllers is desired in order to achieve a better overall control performance. Any resultant improvement in control performance needs, however, to be significant enough to warrant the additional design complexity and higher energy consumption and costs associated with introducing communication channels between controllers. A practical distributed control design aims, therefore, to achieve an acceptable balance between minimising the use of communication between controllers and maximising the system-wide performance. In this article, a new approach to the problem of synthesising stabilising distributed control laws for discrete-time linear systems that balances performance and communication is presented. The approach employs a supervisory agent that, periodically albeit not necessarily at every sampling instant, solves an optimisation problem in order to synthesise a stabilising state feedback control law for the system. The online optimisation problem, which maximises sparsity of the control law while minimising an infinite-horizon performance cost, is formulated as a bilinear matrix inequality (BMI) problem; subsequently, it is then relaxed to a linear matrix inequality (LMI) problem, and (i) convergence to a solution as well as (ii) that early termination guarantees a feasible (but suboptimal) control law are proved. Stability of the closed-loop system under what is a switched control law is guaranteed by the inclusion of dwell-time constraints in the LMI problem. Finally, the efficacy of the approach is demonstrated through numerical simulation examples.

This thesis is concerned with the dissipative control and filtering problems of singular systems. Four classes of singular systems are considered: delay-free singular systems, singular systems with constant time-delay, uncertain singular systems with time-varying delay and sensor failures, and singular Markovian jump systems with actuator failures.
\nFor delay-free singular systems, the system augmentation approach is employed to study the dissipative control and filtering problems. First, the approach is used to solve the dissipative control problem by static output-feedback for standard state-space systems which are the special cases of singular systems. For a continuous-time standard state-space system, the closed-loop system is represented in an augmented system form. Based on the augmented system, a necessary and sufficient dissipativity condition is proposed, which decouples the Lyapunov matrix and controller matrix. To further separate the Lyapunov matrix and the system matrices, an equivalent condition is obtained by introducing some slack matrices. Then, a necessary and sufficient condition for the existence of a static output-feedback controller is proposed, and an iterative algorithm is given to solve the condition. For discrete-time singular systems, by giving an equivalent representation of the solution set, a necessary and sufficient dissipativity condition is proposed in terms of strict linear matrix inequality (LMI) which can be easily solved by standard commercial software. Then a state-feedback controller design method is given based on the augmentation system approach. The method is extended to the static output-feedback control problem and the reduced-order dissipative filtering problem.
\nFor continuous-time singular time-delay systems, the problem of state-feedback dissipative control is considered. An improved delay-dependent dissipativity condition in terms of LMIs is established by employing the delay-partitioning technique, which guarantees a singular system to be admissible and dissipative. Based on this, a delay-dependent sufficient condition for the existence of a state-feedback controller is proposed to guarantee the admissibility and dissipativity of the closed-loop system. In addition to delay-dependence, the obtained results are also dependent on the level of dissipativity. Moreover, the results obtained unify existing results on H∞ performance analysis and passivity analysis for singular systems.
\nFor discrete-time singular systems with polytopic uncertainties, time-varying delay and sensor failures, the problem of robust reliable dissipative filtering is considered. The filter is designed by the reciprocally convex approach such that the filtering error singular system is admissible and strictly (Q, S, R)-dissipative. For singular systems with time-varying delay and sensor failures, a sufficient condition of reliable dissipative analysis is obtained in terms of LMIs. Then the result is extended to the uncertain case by introducing some variables to decouple the Lyapunov matrices and the filtering error system matrices. Moreover, a desired filter for uncertain singular systems with time-varying delay and sensor failures is obtained by solving a set of LMIs.
\nFor continuous-time singular Markovian jump systems with actuator failures, the problem of reliable dissipative control is addressed. Attention is focused on the state-feedback controller design method such that the closed-loop system is admissible and strictly (Q, S, R)-dissipative. A sufficient condition is obtained in terms of strict LMIs. Moreover, the results obtained unify existing results on H∞control and passive control on singular Markovian jump systems.

While the asymptotic stability of positive linear systems in the presence of bounded time delays has been thoroughly investigated, the theory for nonlinear positive systems is considerably less well-developed. This paper presents a set of conditions for establishing delay-independent stability and bounding the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case. Specifically, when the time delays have a known upper bound, we derive necessary and sufficient conditions for exponential stability of (a) continuous-time positive systems whose vector fields are homogeneous and cooperative, and (b) discrete-time positive systems whose vector fields are homogeneous and order preserving. We then present explicit expressions that allow us to quantify the impact of delays on the decay rate and show that the best decay rate of positive linear systems that our bounds provide can be found via convex optimization. Finally, we extend the results to general linear systems with time-varying delays.

In this paper, we consider stabilization and reconstruction of sampled-data Boolean control networks (BCNs) under noisy sampling interval. A sampled-data BCN under noisy sampling interval is first converted into a probabilistic Boolean network (PBN). We then obtain some necessary and sufficient conditions for global stochastic stability of the considered sampled-data BCN under two types of noisy sampling intervals. However, in analyzing the stochastic stability of large-scale sampled-data BCNs under noisy sampling interval, using the above necessary and sufficient conditions leads to huge computational cost. Therefore, for a large-scale sampled-data BCN, we have to transform it into a size-reduced probabilistic logical network. Then by studying the stochastic stability of the probabilistic logical network, some sufficient conditions for global stochastic stability of the large-scale sampled-data BCN are obtained. Moreover, based on the given steady-state probabilities of the transformed PBN, the reconstruction problem of sampled-data BCNs under noisy sampling interval can be well solved as a linear programming (LP) problem.

Stability analysis and control design of LTI discrete-time systems by the direct use of time series data