Continuous-time LTI Systems Research Articles

Design of the Impulsive Goodwin's Oscillator in 1-cycle

This paper presents a systematic approach to design a hybrid oscillator that admits an orbitally stable periodic solution of a certain type with pre-defined parameters. The parsimonious structure of the Impulsive Goodwin's oscillator (IGO) is selected for the implementation due to its well-researched rich nonlinear dynamics. The IGO is a feedback interconnection of a positive third-order continuous-time LTI system and a nonlinear frequency and amplitude impulsive modulator. A design algorithm based on solving a bilinear matrix inequality is proposed yielding the slope values of the modulation functions that guarantee stability of the fixed point defining the designed periodic solution. Further, assuming Hill function parameterizaton of the pulse-modulated feedback, the parameters of those rendering the desired stationary properties are calculated. The character of perturbed solutions in vicinity of the fixed point is controlled through localization of the multipliers. The proposed design approach is illustrated by a numerical example. Bifurcation analysis of the resulting oscillator is performed to explore the nonlinear phenomena in vicinity of the designed dynamics.

Lower Bound Analysis of Lp+ Induced Norm for LTI Systems

In this paper, we focus on the lower bounds of the Lp (p ∈ [1,∞), p = ∞) induced norms of continuous-time LTI systems where input signals are restricted to be nonnegative. This induced norm, called the Lp+ induced norm, is particularly useful for the stability analysis of nonlinear feedback systems constructed from linear systems and static nonlinearities where the nonlinearities provide only nonnegative signals for the case p = 2. To have deeper understanding on the Lp+ induced norm, we analyze its lower bounds with respect to the standard Lp induced norm in this paper. As the main result, we show that the Lp+ induced norm of an LTI system cannot be smaller than the Lp induced norm scaled by 2(1-p)/p for ∈ [1,∞) (scaled by 2−1 for p = ∞). On the other hand, in the case where p = 2, we further propose a method to compute better (larger) lower bounds for single-input systems via reduction of the lower bound analysis problem into a semi-infinite programming problem. The effectiveness of the lower bound computation method, together with an upper bound computation method proposed in our preceding paper, is illustrated by numerical examples.

Identifying Resonant Poles by Visual Inspection of Pole-Zero Plots

We derive the s- and z-plane pole regions for continuous-time and discrete-time LTI systems to yield resonance. In this way, resonance can be identified by visual inspection of the pole-zero plot, without the need for calculations.

Modulation-Function-Based Data-Driven Design of Fault Detection Systems for Continuous-Time LTI Systems

In this paper, a data-driven or model-free approach is presented to design a fault detection system of continuous-time linear time-invariant (LTI) systems based on input and output data in the time domain. The main idea is to directly identify the subspaces and their related matrices relevant for parity-space-based residual generation based on a modulated output equation by use of modulation functions and their properties. Therefore, the explicit model identification of the process for a model-based approach in a conventional two-step procedure can be avoided saving design effort especially for large-scale systems. A simulation of the resulting fault detection system is provided showing the effectiveness of the design approach.

Measure of quality of finite-dimensional linear systems: A frame-theoretic view

Abstract A measure of quality of a control system is a quantitative extension of the classical binary notion of controllability. In this article we study the quality of linear control systems from a frame-theoretic perspective. We demonstrate that all LTI systems naturally generate a frame on their state space, and that three standard measures of quality involving the trace, minimum eigenvalue, and the determinant of the controllability Gramian achieve their optimum values when this generated frame is tight. Motivated by this, and in view of some recent developments in frame-theoretic signal processing, we propose a natural measure of quality for continuous time LTI systems based on a measure of tightness of the frame generated by it and then discuss some properties of this frame-theoretic measure of quality.

A Generalized Minimum Phase Property for Finite-Dimensional Continuous-Time MIMO LTI Systems with Additive Disturbances

In this paper, we further generalize the definition of the extended zero dynamics to finite-dimensional continuous-time MIMO LTI systems with additive disturbances; and then introduce the concept of minimum phase for this class of systems. We show that the extended zero dynamics is invariant under the application of dynamic extension to its input, and therefore, a minimum phase system remains minimum phase after a finite number of steps of dynamic extension. We further introduce the extended zero dynamics canonical form for square MIMO LTI systems with uniform vector relative degree. We prove that a system is minimum phase according to our definition if its zero dynamics is asymptotically stable. The converse of the statement holds if the system is stabilizable from the control input.The objective of this research is to solve the model reference robust adaptive control problem for finite-dimensional continuous-time square MIMO LTI systems that is minimum phase according to the generalized definition using an appropriately vectorized version of Pan and Başar (2000), and results here constitute important building blocks. In a subsequent paper, Başar and Pan (2019), we further connect the dots: starting with a square MIMO LTI system that is minimum phase with respect to admissible initial conditions and admissible disturbance waveforms, we must obtain a true system representation that admits both the extended zero dynamics canonical form representation and the strict observer canonical form representation. Toward this end, we need to be able to extend the given system to one with uniform vector relative degree and with uniform observability indices without changing its minimum phase property, and this has been done in Başar and Pan (2019), fully resolving this issue.

Masking attack for sampled‐data systems via input redundancy

This study discovers a new vulnerability of cyber-physical systems to malicious attack. It arises when the physical plant, that is modelled as a continuous-time LTI system, is controlled by a digital controller with periodic sampling and actuation. In the sampled-data framework, most anomaly detectors monitor the plant's output only at discrete time instants. Consequently, abnormal behaviour between sampling instants cannot be detected if output behaves normally at every sampling instant. This implies that if an actuator attack drives the plant's state to pass through the kernel of the output matrix at each sensing time, then the attack compromises the system while remaining stealthy. It is shown that this type of attack always exists when the sampled-data system has an input redundancy, i.e. the number of inputs being larger than that of outputs and/or the sampling rate of the actuators being higher than that of the sensors. Simulation results for the X-38 vehicle and other numerical examples illustrate this new attack strategy may result in disastrous consequences.

Necessary and Sufficient Bit Rate Conditions to Stabilize a Scalar Continuous-Time LTI System Based on Event Triggering

This paper considers a scalar continuous-time linear time-invariant system, whose feedback signal is transmitted through a communication network. Such a network has only finite bit rate and suffers from transmission delay which is characterized by both lower and upper delay bounds. The concerned system implements event-triggering strategies, i.e., only when certain events are triggered, the system samples and transmits feedback signals. This paper first derives some lower bounds on the feedback bit rate required to achieve the input-to-state stability under arbitrary event-triggering strategies. Then this paper proposes some constructive methods to design the event-triggering strategy and the controller, and can achieve the input-to-state stability at a bit rate being arbitrarily close to these obtained lower bit rate bounds. Moreover, this paper proves that the stabilizing bit rates under the proposed event-triggering strategies can be strictly lower than the stabilizing bit rate required by any time-triggering strategy. This bit rate superiority comes from the fact that under event triggering, the state information can be freely extracted from the receive time instants of feedback packets without consuming any bit rate. Simulations are done to demonstrate the bit rate superiority of the proposed event-triggering strategies.

Direct Adaptive Optimal Control for Uncertain Continuous-Time LTI Systems Without Persistence of Excitation

This brief presents a novel direct adaptive optimal controller design for uncertain continuous-time linear time-invariant systems. The optimal gain parameter, obtained from the Riccati equation, is continuously estimated without using knowledge of the system dynamics, rather information rich past, and current data along the system trajectory is used for parameter estimation. This approach guarantees parameter convergence to the close neighborhood of the optimal controller by relaxing the restrictive persistence of excitation condition, typically required for achieving parameter convergence in approximate/adaptive optimal control methods. A Lyapunov-based analysis establishes the uniformly ultimately bounded stability of the designed controller. Further a simulation example demonstrates the efficacy of the proposed result.

New Stability Tests for Discretized Fractional-Order Systems Using the Al-Alaoui and Tustin Operators

This paper provides new results on a stable discretization of commensurate fractional-order continuous-time LTI systems using the Al-Alaoui and Tustin discretization methods. New, graphical, and analytical stability/instability conditions are given for discrete-time systems obtained by means of the Al-Alaoui discretization scheme. On this basis, an analytically driven stability condition for discrete-time systems using the Tustin-based approach is presented. Finally, the stability of discrete-time systems obtained by finite-length approximation of the Al-Alaoui and Tustin operators are discussed. Simulation experiments confirm the effectiveness of the introduced stability tests.

Continuous time LTI systems under lossless positive real transformations: open-loop balanced representation and truncated reduced-order models

ABSTRACTIn this paper, new results on the open-loop balanced representation of continuous time linear time-invariant systems are reported. More particularly, the effect of lossless positive real transformations on open-loop balanced representations is investigated with specific attention to the problem of model order reduction. The properties of systems where a lossless positive real transformation has been applied are discussed showing that, if the original system is open-loop balanced, the resulting transformed system is still open-loop balanced. Furthermore, the singular values of the transformed system are related to those of the original one. These results allow to derive a model order reduction strategy for this class of systems that leads to a consistent decrease of the numerical complexity. The proposed approach reveals to be of particular interest for the design of reduced-order systems with specific amplitude responses, including analog multiband filters.

Linear Quadratic Optimal Control of Continuous-Time LTI Systems With Random Input Gains

This note studies the linear quadratic (LQ) optimal control of continuous-time linear time-invariant systems with random gains imposed on the input channels. We start from the indefinite LQ problem, in which the cost weighting matrix can be indefinite. The definite LQ problem is discussed as a special case. The main novelty originates from the point of view that in networked control, designing the channels and controller jointly often leads to an easier problem and achieves better performance than designing them separately. Specifically, we formulate the LQ problem as a channel/controller co-design problem assuming that the channel capacities can be allocated among the input channels subject to an overall capacity constraint. Necessary and sufficient conditions are obtained for the well-posedness and the attainability of the indefinite LQ problem under a given channel capacity allocation satisfying the stabilization requirement. The optimal controller is given by a linear state feedback associated with the mean-square stabilizing solution of a modified algebraic Riccati equation.

Filtering Continuous-Time Linear Systems With Time-Varying Measurement Delay

We consider the filtering problem of LTI continuous-time systems with known and bounded measurement delays. The aim of the technical note is the design of a finite-dimensional sub-optimal filter whose performance in terms of the estimation error is comparable to optimal infinite-dimensional approaches. We show that the proposed approach allows for a precise characterization of the relationship between measurement delay and the covariance of the estimation error. In the time-varying case no restrictive hypotheses on the delay function are needed. The proposed filter can therefore be applied to delay functions for which traditional infinite-dimensional approaches cannot be straightforwardly applied.

Optimal input design for continuous-time system identification: application to fractional systems

An optimal input design framework for continuous-time system identification using fractional models is presented. This framework extends some existing results in discrete-time systems. The problem is formulated in convex finite dimensional form. Input spectrum decomposition in Laguerre polynomial basis is applied and an LMI solution is proposed. This method permits to synthesize optimal input spectrum for LTI continuous-time systems including fractional ones.

Design of quantised dynamic output feedback for decentralised ℋ ∞ control systems

The design of quantised dynamic output feedback for decentralised ℋ ∞ control systems is considered with multi-input and multi-output. It is assumed that a decentralised dynamic output feedback has been designed for a decentralised continuous-time LTI system, so that the closed-loop system is stable and a desired ℋ ∞ disturbance attenuation level is achieved, and that each channel's measurement outputs are quantised before they are passed to the local controller. We propose a local-output-dependent strategy for updating the quantisers' parameters, so that the overall closed-loop system is asymptotically stable and achieves the same ℋ ∞ disturbance attenuation level. Both the pre-designed controllers and the quantisers' parameters are constructed in a decentralised manner, depending on local measurement outputs.

Decentralized design of interconnected H∞ feedback control systems with quantized signals

In this paper, we consider the design of interconnected H∞ feedback control systems with quantized signals. We assume that a decentralized dynamic output feedback has been designed for an interconnected continuous-time LTI system so that the closed-loop system is stable and a desired H∞ disturbance attenuation level is achieved, and that the subsystem measurement outputs are quantized before they are passed to the local controllers. We propose a local-output-dependent strategy for updating the parameters of the quantizers, so that the overall closed-loop system is asymptotically stable and achieves the same H∞ disturbance attenuation level. Both the pre-designed controllers and the parameters of the quantizers are constructed in a decentralized manner, depending on local measurement outputs.

Minimum-time trajectory generation for constrained linear systems using flatness and B-splines

In this article, we discuss minimum-time trajectory generation for input-and-state constrained continuous-time LTI systems in the light of the notion of flatness and B-splines parametrisation. Flat systems have the useful property that the input and the state trajectories can be completely characterised by the (so-called) flat output. We propose a splines parametrisation for the flat output, and the corresponding parametrisations for the performance output, the states and the inputs. Using this parametrisation the problem of minimum-time constrained trajectory planning is cast into a feasibility-search problem in the splines control-point space, in which the constraint region is characterised by a polytope. A close approximation of the minimum-time trajectory is obtained by systematically searching the end-time that makes the constraint polytope to be minimally feasible.

Flatness-based Minimum-time Trajectory Generation for Constrained Linear Systems Using B-Splines

AbstractIn this paper we discuss minimum-time trajectory generation for input-and-state constrained continuous-time LTI systems in the light of the notion of flatness and B-splines parameterisation. Flat systems have the useful property that the input and the state trajectories can be completely characterised by the so-called flat output. We propose a splines parameterisation for the flat output, and the corresponding parameterisations for the performance output, the states, and the inputs. Using this parameterisation the problem of minimum-time constrained trajectory planning is cast into a feasibility-search problem in the splines control-point space, in which the constraint region is characterised by a polytope. A close approximation of the minimum-time trajectory is obtained by systematically searching the end-time that makes the constraint polytope to be minimally feasible.

System Identification With Sparse Coprime Sensing

Given a continuous time LTI system with impulse response h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> (t), it is shown that the uniformly spaced samples h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> (nT) can be identified for any chosen spacing T by using an impulse train input with an arbitrarily small rate 1/NT and sampling the system output with an arbitrarily small rate 1/MT , provided M and N are coprime. This idea, referred to here as the sparse coprime sensing method for system identification, is closely related to well known results in multirate signal processing. It is shown that the problem can be related to the identification of a decimation filter from input-output measurements. It is also shown that the problem is equivalent to the identification of a discrete time N × M LTI system from a knowledge of the full rate input and output vector sequences.

Quantizer design for interconnected feedback control systems

In this paper, we consider the design of interconnected H-infinity feedback control systems with quantized signals. We assume that a decentralized static output feedback has been designed for an interconnected continuous-time LTI system so that the closed-loop system is stable and a desired H-infinity disturbance attenuation level is achieved, and that the subsystems’ measurement outputs are quantized before they are passed to the local controller. We propose a localoutput-dependent strategy for updating the quantizers’ parameters, so that the overall closed-loop system is asymptotically stable and achieves the same H-infinity disturbance attenuation level. Both the pre-designed controllers and the quantizers’ parameters are constructed in a decentralized manner, depending on local information.