Control Of Discrete-time Linear Systems Research Articles

On Construction of A Control Operator Introduced To ECGM Algorithm for Solving Discrete-Time Linear Quadratic Regulator Control Systems with Delay-I

In this paper, we constructed a control operator sequel to an earlier constructed control operator in one of our papers which enables an Extended Conjugate Gradient Method (ECGM) to be employed in solving discrete time linear quadratic regulator problems with delay parameter in the state variable. The construction of the control operator places scalar linear delay problems of the type within the class of problems that can be solved with the ECGM and it is aimed at reducing the rigours faced in using the classical methods in solving this class of problem. More so, the authors of this paper desire that, the application of this control operator will further improve the results of the ECGM as well as increasing the variant approaches used in solving the said class of optimal control problem.

Networked control of discrete-time linear systems over lossy digital communication channels

This article addresses networked control problems for linear time-invariant systems. The insertion of the digital communication network inevitably leads to packet dropout, time delay and quantisation error. Due to data rate limitations, quantisation error is not neglected. In particular, the case where the sensors and controllers are geographically separated and connected via noisy, bandwidth-limited digital communication channels is considered. A fundamental limitation on the data rate of the channel for mean-square stabilisation of the closed-loop system is established. Sufficient conditions for mean-square stabilisation are derived. It is shown that there exists a quantisation, coding and control scheme to stabilise the unstable system over packet dropout communication channels if the data rate is larger than the lower bound proposed in our result. An illustrative example is given to demonstrate the effectiveness of the proposed conditions.

Linear Unbiased Optimal Filter for Discrete-Time Systems with One-Step Random Delays and Inconsecutive Packet Dropouts

This paper is concerned with the linear unbiased minimum variance estimation problem for discrete-time stochastic linear control systems with one-step random delay and inconsecutive packet dropout. A new model is developed to describe the phenomena of the one-step delay and inconsecutive packet dropout by employing a Bernoulli distributed stochastic variable. Based on the model, a recursive linear unbiased optimal filter in the linear minimum variance sense is designed by the method of completing the square. The solution to the linear filter is given by three equations including a Riccati equation, a Lyapunov equation and a simple difference equation. A sufficient condition for the existence of the steady-state filter is given. A simulation shows the effectiveness of the proposed algorithm. DOI: http://dx.doi.org/10.11591/telkomnika.v11i11.3523

Consensus based Approach to the Signal Control of Urban Traffic Networks

The paper presents a signal control strategy of urban traffic networks (UTNs), under which states of UTNs can realize asymptotic stable consensus. We first propose a simulation model of UTNs, where the topology of road network is represented by a directed dynamic graph, and the transfer of underlying traffic flows is modeled by the cell transmission model (CTM). Furthermore, under some assumptions, we can obtain a signal control model of UTNs, which is a discrete-time linear time-invariant control system, and then design a state-feedback control law, under which the proposed control system can realize asymptotic stable consensus. At last, we illustrate basic ideas of our methods by an example.

A Sequential Linear Programming Matrix Method to Insensitive H∞ Output Feedback for Linear Discrete-Time Systems

This paper studies the problem of designing insensitive H∞ output-feedback controllers for linear discrete-time systems. The designed controllers are insensitive to additive/multiplicative controller coefficient variations. An LMI-based procedure, which is a sequential linear programming matrix method (SLPMM), is proposed to solve the considered problem which is a nonconvex problem itself. It is worth mentioning that the nonfragile control design method is adopted to obtain an effective solution for accelerating convergence of SLPMM algorithm due to the fact that a good starting point for the iteration is very important.

Computation of the Equilibrium Point for Discrete-time Linear Stochastic Systems

The linear optimal control problem of linear systems subject to Markov jumps and/or multiplicative noises is considered and the computation of an optimal control strategy is commented. This kind of models has found many applications in engineering and finance as, for instance, in nuclear fission and heat transfer, population models and immunology, portfolio optimization (O. Costa, de Oliviera, Optimal mean-variance control for discrete-time linear systems with Markovian jumps and multiplicative noises, Automatica 48(2012), 304--315), etc., and several results related to the control of these systems have already been derived. The challenge of the applications is that the weighting matrices in the linear quadratic models are assumed to be singular or indefinite. The realization of the optimal control strategy depends on the stabilizing solution of some appropriate systems of Riccati-type coupled equations, i.e. a set of generalized discrete-time algebraic Riccati equations has to be solved. The LMI approach for computing the stabilizing symmetric solution of this system is studied. We construct two new modifications of the standard LMI approach and we show how to apply these new modifications to the investigated problem. Computer realizations of all modifications are compared. Numerical experiments are given where the new LMI modifications are numerically compared. Based on these experiments the main conclusion is that the new LMI modifications are faster than the standard LMI approach.

Controllability of Linear Discrete-Time Systems with Both Delayed States and Delayed Inputs

The controllability issues for discrete-time linear systems with delay in state and control are addressed. By introducing a new concept, the controllability realization index (CRI), the characteristic of controllability is revealed. An easily testable necessary and sufficient condition for the controllability of discrete-time linear systems with state and control delay is established.

Quantized H∞ output feedback control for linear discrete-time systems

This paper investigates the robust H∞ dynamic output feedback control problem for networked control systems (NCSs) with quantized measurements. The measurement losses of the communicated information are considered in an unreliable communication channel. The robust H∞ dynamic output feedback controllers are designed to handle the measurement losses and mitigate the quantization effects such that the resultant closed-loop NCS is mean-square stochastically stable with a prescribed H∞ disturbance attenuation performance. The controller existence conditions can be derived in terms of linear matrix inequalities (LMIs). Finally, an example is provided to illustrate the effectiveness of the proposed approach.

Control of linear discrete-time systems by quantised feedback

In this study the authors, we investigate the problem of designing quantised ℋ∞ feedback control for a class of discrete-time systems with and without bounded uncertainties. The quantiser has arbitrary form. A linear matrix inequality-based method using static and dynamic quantised output-feedback controller are designed to render the closed-loop system asymptotically stable with guaranteed γ-level. The authors illustrate the theoretical developments by numerical simulations.

H∞ control of discrete-time linear systems constrained in state by equality constraints

Abstract In this paper, stabilizing problems in control design are addressed for linear discrete-time systems, reflecting equality constraints tying together some state variables. Based on an enhanced representation of the bounded real lemma for discretetime systems, the existence of a state feedback control for such conditioned stabilization is proven, and an LMI-based design procedure is provided. The control law gain computation method used circumvents generally an ill-conditioned singular design task. The principle, when compared with previously published results, indicates that the proposed method outperforms the existing approaches, guarantees feasibility, and improves the steady-state accuracy of the control. Furthermore, better performance is achieved with essentially reduced design effort. The approach is illustrated on simulation examples, where the validity of the proposed method is demonstrated using one state equality constraint.

Fractional discrete-time linear control systems with initialisation

Linear control systems with fractional differences and finite memory are considered. The evaluation of trajectories of such systems is studied. The extension of the classical notion of observability and controllability is given and developed. The necessary and sufficient conditions for observability and controllability are proved.

Adaptive Control with Guaranteed Contraction Rate for Systems with Actuator Saturation

The problem of adaptive control of linear discrete-time systems with actuator saturation and unknown parameters is investigated. A novel optimal control method is presented first for linear systems with known parameters and constant actuator saturation by introducing a Lyapunov function and a performance cost function that are both dependent on a contraction rate parameter. Based on the obtained guaranteed contraction-rate control method, an adaptive control algorithm is derived for systems containing unknown system parameters and time-varying actuator saturation. To show that the closed-loop system is stable and that the adaptive control algorithm is convergent, the Lyapunov function is supplemented by an additional part defined by the trace of a quadratic function of the controller gain. The effectiveness and potential of the presented method is demonstrated by a numerical example.

Controllability Analysis of Linear Discrete Time Systems with Time Delay in State

The controllability issues for linear discrete‐time systems with delay in state are addressed. By introducing a new concept, the minimum controllability realization index (MinCRI), the characteristic of controllability is revealed. It is proved that the MinCRI of a system with state delay exists and is finite. Based on this result, a necessary and sufficient condition for the controllability of discrete‐time linear systems with state delay is established.

筋電位の電気力学的遅延を用いた零位相追従型マンマシンインターフェースの開発

This study aims to develop a man-machine interface which realizes the perfect tracking to human motions. With such an interface, a user may operate machines with improved maneuvering feel. The perfect tracking can be realized by utilizing Zero Phase Error Tracking Control (ZPETC) to eliminate or reduce phase lags including mechanical and communication delays in systems. In general, ZPETC however requires future information as reference signals. We propose to utilize a time difference from arising electromyogram (EMG) signals in muscles until appearing the corresponding actual human motion. The time difference is called Electro Mechanical Delay (EMD). Hence, if human motions can be estimated from electromyogram signals, we can obtain the future information on the human motions by EMD. Our system focuses on wrist motion as commands toward a system. Utilizing EMD, the wrist angle is estimated from EMG signals by a nonlinear delayed ARX (NDARX) model. A discrete-time Model Following (MF) controller is used for letting the system track the wrist motion. Consequently, this demonstrates the possibility that the controlled system can move tracking the wrist motion with no or less lag with respect to the practical human motion. Friction compensation plays an important role from the implementation point of view, because ZPETC is a feedforward controller for discrete-time linear systems but the real system is affected by friction. The effectiveness is verified through numerical simulations and experiments comparing tracking performance using only a PID controller with the proposed control system in terms of cross-correlation analysis.

Control of linear systems with non-convex constraints. Mixed integer formulations, MPC design and interpolation alternatives

The paper deals with the control of linear discrete-time systems in presence of non-convex state-space constraints, understood as obstacles for the closed-loop trajectories. These interdicted regions can include the origin and as a consequence, the natural limit behavior has to be replaced by a new admissible equilibrium point, possibly by activating a subset of constraints. It will be shown that a predictive control strategy can deal, by its generic formulation, with such problems and several refinements will be pointed out with respect to the dual-mode principles and the numerical formulations of the optimization problems. Furthermore, the second goal of the paper is to present an interpolation-based procedure as a reduced-complexity alternative to the classical receding horizon optimization. The basic elements for the establishment of such a control scheme are the boundary of the closed convex sets in the state-space and the associated vertex controller.

A study on the state set invariance for a class of discrete-time linear control systems

This paper considers the state set invariance for a class of linear control systems over the discrete time horizon. Given a state set and a control set, the author proposes the methodology to design a state feedback controller and its corresponding maximum invariant subset of the state set. The corresponding invariant subset means that with the controller any state starting from the subset remains in the subset at all times. Fixing the controller, the necessary and sufficient condition for the maximum invariant subset to be the state set itself is first derived. For the systems not satisfying the condition, the author characterizes the corresponding maximum invariant subset based on the spectral analysis of matrices. Numerical examples for different cases are given to illustrate the proposed computation techniques.

Observer-Based Reduced Order Controller Design for the Stabilization of Large Scale Linear Discrete-Time Control Systems

This paper discusses the design of observer-based reduced order controllers for the stabilization of large scale linear discrete-time control systems. This design is carried out via deriving a reduced-order model for the given linear plant using the dominant state of the linear plant. Using this reduced-order linear model, sufficient conditions are derived for the design of observer-based reduced order controllers. A separation principle has been established in this paper which demonstrates that the observer poles and controller poles can be separated and hence the pole-placement problem and observer design are independent of each other.

Optimal mean–variance control for discrete-time linear systems with Markovian jumps and multiplicative noises

In this paper, we consider the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noises under two criteria. The first one is an unconstrained mean–variance trade-off performance criterion along the time, and the second one is a minimum variance criterion along the time with constraints on the expected output. We present explicit conditions for the existence of an optimal control strategy for the problems, generalizing previous results in the literature. We conclude the paper by presenting a numerical example of a multi-period portfolio selection problem with regime switching in which it is desired to minimize the sum of the variances of the portfolio along the time under the restriction of keeping the expected value of the portfolio greater than some minimum values specified by the investor.

Optimal Reset Law Design and Its Application to Transient Response Improvement of HDD Systems

This brief investigates optimal reset law design of reset control systems with fixed reset time instants. First, closed-loop reset control systems are changed to discrete-time linear control systems with reset values as control inputs. Then the optimal reset law design problems are transferred to linear quadratic regulation problems and the optimal reset laws are then obtained by solving algebraic Riccati equations. As an application, the proposed technique is applied to track-seeking control of a kind of hard disk drive systems. Simulations show that the transient response is improved significantly. Especially, the proposed long-span seeking reset control is almost bang-bang, but the design process is much simpler.

Supervisory Control and Measurement Scheduling for Discrete-Time Linear Systems

The problem of jointly synthesizing a supervisor, a measurement scheduler, and a feedback controller for discrete-time linear systems is considered. It is shown that open-loop supervisory and scheduling laws are nonconservative for robust exponential stability and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> type performance requirements, and that they can be obtained separately from the feedback controller. All the design conditions are formulated in terms of linear matrix inequalities.