Discrete Time-invariant Systems Research Articles

Parity space-based fault isolation using minimum error minimax probability machine

In this paper, a new parity space-based fault isolation (FI) approach is proposed for linear discrete time-invariant systems. The main contribution lies in the integrated design of FI system and fault isolability properties without any assumptions on the probability distribution of unknown inputs. First, a fault isolability evaluation index is defined as the worst-case probability of correct FI in different faulty scenarios, and a new objective function is proposed for the integrated design of parity space-based residual generator, evaluation logic and fault isolability properties. Based on this, the FI problem is formulated as a bank of optimal multiclass classification problems. Next, using the technique of minimum error minimax probability machine (MEMPM), a group of feasible parity space matrices for residual generation are obtained. Furthermore, the FI method by MEMPM is presented with a desirable trade-off between the fault isolability properties in different faulty scenarios. To show the effectiveness of the proposed method, a satellite attitude control system subject to six faults is considered.

Synthesis of Anisotropic Suboptimal PID Controller for Linear Discrete Time-Invariant System with Scalar Control Input and Measured Output

This paper considers the problem of synthesis of a proportional-integral-derivative control law (PID controller) for a linear discrete time-invariant system with scalar control input and measured output operating under influence of the stochastic disturbances with uncertainty described in terms of the mean anisotropy. The closed-loop system abilities to attenuate the disturbances are quantitatively characterized by the anisotropic norm. Sufficient existence conditions for the anisotropic suboptimal controller that stabilizes the closed-loop system and guarantees that its anisotropic norm is strictly bounded by a given threshold value are derived.

Asymptotic Unknown Input Decoupling Observer for Discrete-Time LTI Systems

This letter addresses a new observer structure and its corresponding design for linear time invariant discrete-time systems affected by unknown inputs (UIs). We show that the performances of standard unknown input observer (UIO) can be enhanced, especially, in the presence of stable invariant zeros with slow dynamics (detectable systems). Under mild condition, the proposed UIO design avoids the drawback of the invariant zeros, at least in a time interval, and allows to arbitrary place the eigenvalues during that interval. The main idea is to relax the exact UI decoupling condition for an asymptotic decoupling condition. The convergence analysis is analyzed using the Lyapunov theory and the asymptotic convergence conditions of the state estimation error are expressed in linear matrix inequality formalism.

Prognosis of uncertain linear time-invariant discrete systems using unknown input interval observer

In this paper, a model-based prognosis method where the degradation cannot be directly measured but only detectable through the drift of a model parameter is considered. This parameter drift is viewed as an unknown input, whose reconstruction allows the estimation of the degradation state. Model-based prognosis is divided into a filtering step where the current degradation state is estimated, and an uncertainty propagation step where the future degradation state is predicted. During these two steps, model uncertainty and measurement uncertainty are taken into account within the set-membership framework using interval techniques. The filtering step is performed with an interval unknown input observer for linear time-invariant discrete systems. Then, based on a set-membership constraint satisfaction methodology, interval propagation is performed to estimate the bounds that include the future degradation state until some failure threshold is reached, allowing the deduction of the interval including the remaining useful life. In order to demonstrate the efficiency of the proposed model-based prognosis methodology, the degradation of a suspension system is studied.

Frequency domain response under arbitrary excitation for fading memory nonlinear systems

For dynamic systems, the steady-state system response to periodic excitation is well understood for both linear and certain nonlinear system classes. When the excitation is not periodic, however, the measured response will contain both transient and steady-state contributions. For linear systems, these transient contributions have been thoroughly explored, while no equivalent analysis exists for the nonlinear case. In this paper, we derive an expression in the frequency domain for the system response of all discrete time-invariant nonlinear systems which have fading memory, using the Volterra series representation. The expression contains both steady-state and transient contributions at each nonlinear order, revealing a highly structured view of nonlinear system response. For the nonlinear case, the transient expressions at higher nonlinear orders have a more complex structure than those generated by linear systems, which provides valuable insight for systems theory and identification purposes.

Analysis and Design of Actuation–Sensing–Communication Interconnection Structures Toward Secured/Resilient LTI Closed-Loop Systems

This paper considers the analysis and design of resilient/robust decentralized control systems. Specifically, we aim to assess how the pairing of sensors and actuators lead to architectures that are resilient to attacks/hacks for industrial control systems and other complex cyber-physical systems. We consider inherent structural properties such as internal fixed modes of a dynamical system depending on actuation, sensing, and interconnection/communication structure for linear discrete time-invariant dynamical systems. We introduce the notion of a resilient fixed-modes free system that ensures the nonexistence of fixed modes when the actuation–sensing–communication structure is compromised due to attacks by a malicious agent on actuators, sensors, or communication components and natural failures. Also, we provide a graph-theoretical characterization for the resilient structurally fixed modes that enables to capture the nonexistence of resilient fixed modes for almost all possible systems’ realizations. Additionally, we address the minimum actuation–sensing–communication codesign ensuring the nonexistence of resiliently structurally fixed modes, which we show to be NP-hard. Notwithstanding, we identify conditions that are often satisfied in engineering settings and under which the codesign problem is solvable in polynomial-time complexity. Furthermore, we leverage the structural insights and properties to provide a convex optimization method to design the gain for a parameterized system and satisfying the sparsity of a given information pattern. Thus, exploring the interplay between structural and nonstructural systems to ensure their resilience. Finally, the efficacy of the proposed approach is demonstrated on a power grid example.

Discrete Time-Delay Optimal Control Method for Experimental Active Chatter Suppression and Its Closed-Loop Stability Analysis

Abstract Chatter is a self-excited and unstable vibration phenomenon during machining operations, which affects the workpiece surface quality and the production efficiency. Active chatter control has been intensively studied to mitigate chatter and expand the boundary of machining stability. This paper presents a discrete time-delay optimal control method for chatter suppression. A dynamical model incorporating the time-periodic and time-delayed characteristic of active chatter suppression during the milling process is first formulated. Next, the milling system is represented as a discrete linear time-invariant (LTI) system with state-space description through averaging and discretization. An optimal control strategy is then formulated to stabilize unstable cutting states, where the balanced realization method is applied to determine the weighting matrix without trial and error. Finally, a closed-loop stability lobe diagram (CLSLD) is proposed to evaluate the performance of the designed controller based on the proposed method. The CLSLD can provide the stability lobe diagram with control and evaluate the performance and robustness of the controller cross the tested spindle speeds. Through many numerical simulations and experimental studies, it demonstrates that the proposed control method can make the unstable cutting parameters stable with control on, reduce the control force to 21% of traditional weighting matrix selection method by trial and error in simulation, and reduce the amplitude of chatter frequency up to 78.6% in experiment. Hence, the designed controller reduces the performance requirements of actuators during active chatter suppression.

Anisotropy-Based Analysis for the Case of Nonzero Initial Condition

Usually in the context of the anisotropy-based robust performance analysis stochastic systems with zero initial condition are investigated. In this paper we extend this analysis and consider a linear discrete time invariant system under random disturbances and with the nonzero initial condition. In accordance with the basic postulates of the anisotropy-based control theory the disturbance attenuation capabilities of system are quantified by the anisotropic generalized gain which is defined as the largest root mean square gain of the system with respect to a random input and the nonzero initial condition, anisotropy of which is bounded by a given nonnegative parameter a.

Discrete Fourier transform based frequency characteristics of iterative learning control for linear discrete-time systems

For discrete-time iterative learning control systems, the discrete Fourier transform (DFT) is a powerful technique for frequency analysis, and Toeplitz matrices are a typical tool for the system input–output transmission. This paper first exploits z-transform and DFT-based frequency properties for iterative learning control systems and studies the convergence property of a Toeplitz matrix to the power of iteration index. The exploitation exhibits that for the finite-length discrete-time iterative learning control systems, the time-domain convolution theorem for the z-transform and DFT is no longer true, and the Toeplitz matrix to the power of iteration index converges if and only if the identical diagonal element lies in the unit circle. Then, by considering the DFT to a finite-length sequence as a linear transform, it is easy to equivalently reform the input–output equation of linear discrete time-invariant and time-varying ILC systems as an algebraic discrete-frequency equation. Thus the derivative-type (D-type) iterative learning control (ILC) converges in a discrete-frequency domain if and only if it converges in a discrete-time domain. Numerical simulations are carried out to exhibit the validity and effectiveness.

Improving Boundary Level Calculation in Quantized Iterative Learning Control With Encoding and Decoding Mechanism

This paper investigates an iterative learning control for single-input, single-output, and linear time-invariant discrete system. The special design of the learning gain matrix is introduced, where a finite uniform quantizer is incorporated with an encoding and decoding mechanism to realize the zero-error convergence of a tracking problem. Furthermore, the boundary-level calculation is considerably improved using lifting technique and infinity-norm of vectors under this mechanism. Some illustrations of the simulations verify the theoretical results.

Discrete state space systems of fractional order

The present article devotes to the study of linear time invariant discrete fractional order state space systems using a novel approach. First, we replace the conventional Grünwald-Letnikov-type backward difference operator with the equivalent Riemann-Liouville-type nabla difference operator and obtain the system response using variation of constants and discrete Laplace transform methods. Next, we present three different algorithms to construct state transition matrix of the system. Finally, we provide an example to illustrate the applicability of established result.

Discrete state space systems of fractional order

The present article devotes to the study of linear time invariant discrete fractional order state space systems using a novel approach. First, we replace the conventional Grunwald-Letnikov-type backward difference operator with the equivalent Riemann-Liouville-type nabla difference operator and obtain the system response using variation of constants and discrete Laplace transform methods. Next, we present three different algorithms to construct state transition matrix of the system. Finally, we provide an example to illustrate the applicability of established result.

On the Control Design for Linear Time–Invariant Systems with Moments Constraints of Disturbances in Anisotropy–based Theory

Abstract In this paper, the design procedure of anisotropic suboptimal dynamic output regulator for linear discrete time-invariant (LDTI) system is considered. The mean anisotropy level of exogenous disturbance is bounded by the certain positive value. Additional constraints of exogenous disturbance are connected with the first and the second moments. The anisotropic suboptimal regulator design problem is formulated in terms of solvability of linear matrix inequalities (LMI) under convex constraints.

Event-triggered MPC design for distributed systems with network communications

This paper deals with the communication problem in the distributed system, considering the limited battery power in the wireless network and redundant transmission among nodes. We design an event-triggered model predictive control U+0028 ET-MPC U+0029 strategy to reduce the unnecessary communication while promising the system performance. On one hand, for a linear discrete time-invariant system, a triggering condition is derived based on the Lyapunov stability. Here, in order to further reduce the communication rate, we enforce a triggering condition only when the Lyapunov function will exceed its value at the last triggered time, but an average decrease is guaranteed. On the other hand, the feasibility is ensured by minimizing and optimizing the terminal constrained set between the maximal control invariant set and the target terminal set. Finally, we provide a simulation to verify the theoretical results. It U+02BC s shown that the proposed strategy achieves a good trade-off between the closed-loop system performance and communication rate.

Adaptive State Observer Synthesis Based On Instrumental Variables Method

The present paper presents non-recurrent adaptive observation algorithm for SISO linear time-invariant discrete systems. The algorithm is based on the instrumental variables method and the adaptive state observer estimates the parameters, the initial and the current state vectors of discrete systems. The algorithm workability is proved by using simulation data in the MATLAB/Simulink environment.

Suboptimal control of bunches of trajectories of discrete deterministic automaton time-invariant systems

The optimal control problem for discrete deterministic time-invariant automaton systems under parametric uncertainty is considered. The changes in the state (switching) of the system are described by a recurrence equation. The switching times and their number are not specified in advance—they are found by optimization. The initial state of the system is not known exactly. For this reason, the problems of finding the optimal on the average and the optimal guaranteeing (minimax) controls of bunches of trajectories are formulated. It is proposed to solve these problems using the separation principle, which assumes that the bunch is controlled using the control that is optimal for a specially chosen individual trajectory in the bunch. Generally, this control is suboptimal. Algorithms for the synthesis of suboptimal control of bunches are developed. It is proved that, in the linear-quadratic time-invariant problems, the suboptimal bunch control is optimal on the average or is the optimal minimax control, depending on the problem being solved.

Development and performance evaluation of an infinite horizon LQ optimal tracker

The paper presents an infinite horizon LQ optimal tracking control solution (LQ tracker) for discrete time linear time invariant systems. The reference preview need is reduced to only two steps irrespective of the type of reference signal making real-time implementation an achievable goal. A rigorous proof of optimality is provided for a set of infinite horizon reference commands which includes the linear combination of constant and exponentially bounded signals. Dissipativity, finite gain and l1 performance of the controlled system are also evaluated. The behaviour of the proposed LQ tracker and its previously published sub-optimal version with one-step preview is demonstrated in conjunction with an application example. Their performances are compared to those of alternative solutions including set point control and model predictive control considering aspects of practical application. Finally, it is concluded that the proposed rigorous solution of the infinite horizon tracking problem and its sub-optimal version are real-time realizable and perform advantageously in some aspects compared to other solutions.

Output Feedback Stable Stochastic Predictive Control With Hard Control Constraints

We present a stochastic predictive controller for discrete time linear time invariant systems under incomplete state information. Our approach is based on a suitable choice of control policies, stability constraints, and employment of a Kalman filter to estimate the states of the system from incomplete and corrupt observations. We demonstrate that this approach yields a computationally tractable problem that should be solved online periodically, and that the resulting closed loop system is mean-square bounded for any positive bound on the control actions. Our results allow one to tackle the largest class of linear time invariant systems known to be amenable to stochastic stabilization under bounded control actions via output feedback stochastic predictive control.

On upper estimate of anisotropic norm of uncertain system with application to stochastic robust control

ABSTRACTThis paper considers a stable linear discrete time-invariant system with linear-fractional norm-bounded uncertainty. An estimate of the anisotropic norm of this system is derived. It is shown that the anisotropic norm is bounded above if a system of inequalities of certain kind is solvable. This system consists of two linear matrix inequalities and inequality with respect to the determinant of a positive definite matrix. By minimising the squared threshold value under constraints in form of the system of matrix inequalities one can obtain the upper estimate of the anisotropic norm of the considered system. The introduced analysis result is directly applied to solve the problem of synthesis of state-feedback robust anisotropic controller.

On the lower bound of the anisotropic norm of the linear stochastic system

The anisotropic norm of the linear discrete time invariant system is the measure of the system output sensitivity to a random input with the mean anisotropy bounded by some nonnegative value a. The mean anisotropy characterizes the degree of predictability of the stochastic signal. The anisotropic norm of a system is the induced norm with the extreme cases of H2 norm and H∞ norm, respectively, under a → 0 and a → ∞. The lower bound of the anisotropic norm of the linear discrete time invariant system was established in the present paper using the methods of linear algebra.