LTI Systems Research Articles

Design of maximum-stability PID controllers for LTI systems based on a stabilizing-set construction method

BackgroundThe PID algorithm has been widely used in control of chemical processes. A maximum-stability PID controller can provide superior stability robustness towards plant's variations and maximum exponential decay rate for disturbance rejection. However, design of maximum-stability PID controller is a min-max optimization problem and an effective method is still lack in the literature. MethodsBased on the characterization of stabilizing PID controller set, an efficient algorithm is developed to test if a plant is σ-stabilizable, where σ is abscissa or stability degree of the Hurwitz stable closed-loop characteristic polynomial. This algorithm is then used along with a bisection strategy to find a σ-interval [σε*,σε*+ε] which contains the maximum stability degree σ* for a specified ε, and the PID controller parameter set for achieving the stability degree σε*. Significant findingsThis paper has presented a systematic and efficient approach to design PID controllers with maximum degree of stability. The principal results include: (i) an improved theorem is presented for identifying stabilizing kp-intervals such that unnecessary computations are avoided; (ii) a simple yet effective method has been adopted to provide a non-conservative interval of σ which facilitates the bisectional branch-and-bound operation; (iii) the design procedure does not involve the actual construction of stabilizing PID controller sets thus renders its efficiency.

A graph-oriented approach to address generically flat outputs in structured LTI discrete-time systems

This paper addresses difference flatness for structured LTI discrete-time systems. Two forms of necessary and sufficient conditions for an output to be a structural flat output are given. First, a preliminary result algebraically defines a flat output in terms of invariant zeros regardless whether an LTI system is structured or not. Next, the conditions are expressed in terms of graphical conditions to define a structural flat output. Checking for the graphical conditions calls for algorithms that have polynomial-time complexity and that are commonly used for digraphs. The tractability of the conditions is illustrated on several examples.

Target Controllability of Networked LTI Systems

A generalized-eigenvector-based condition is derived for the target controllability, which is necessary and sufficient. It is then applied to networked MIMO LTI systems with some lower-dimensional conditions established. These conditions are easily-verified, which explicitly show how the node-system dynamics, network topology, inner couplings, target-node positions, and external control inputs affect the target controllability of the network. Then, the target controllability of multi-agent systems is revisited. The omission of the criterion given in [1] is revealed. As a remedy, a modified criterion is suggested. Several examples show the effectiveness of the conditions.

Robust Control of a Multi-Axis Shape Memory Alloy-Driven Soft Manipulator

Control of soft robotic manipulators remains a challenge for designs with advanced capabilities and novel actuation. Two significant limitations are multi-axis, three-dimensional motion of soft bodies alongside actuator dynamics and constraints, both of which are present in shape-memory-alloy (SMA)-powered soft robots. This article addresses both concerns with a robust feedback control scheme, demonstrating state tracking control for a soft robot manipulator of this type. Our controller uses a static beam bending model to approximate the soft limb as an LTI system, alongside a singular-value-decomposition compensator approach to decouple the multi-axial motion and an anti-windup element for the actuator saturation. We prove stability and verify robustness of our controller, with robustness intended to account for the unmodeled dynamics. Our implementation is verified in hardware trajectory tracking tests of a soft SMA-powered limb with accurate, promising results for future multi-limbed robots.

A note on invertibility and relative degree of MIMO LTI systems

The property of system invertibility is important both from a theoretical standpoint and an important consideration in left and right inversion problems that deal with input estimation and tracking control respectively. Nuances associated with the notion of relative degree in multi-input, multi-output (MIMO) systems on account of input–output delays pose challenges in system inversion problems and are more acute in the case of non-square systems. In this note, using the LTI discrete-time MIMO systems framework, we review existing notions of relative degree and develop new definitions and generalizations for relative degree to help address some of the gaps in system inversion problems. In doing so, we also show through simple but elegant results that existence of well-defined relative degree for an LTI MIMO system implies its invertibility. Furthermore, we also highlight the utility of these ideas in practical tracking control and input estimation situations and problems.

Uncertain AoI in stochastic optimal control of constrained LTI systems

Abstract This paper addresses finite-time horizon optimal control of control structures with shared communication network. To cope with the uncertainties, induced by network imperfections and exogenous disturbances at the same time, an optimization-based control scheme is proposed. It uses a disturbance feedback and the Age of Information (AoI), a receiver-based measure of communication delays, as central aspects. The disturbance feedback is an extension of the control law used for balanced stochastic optimal control. Balanced optimality is understood as a compromise between minimizing expected deviations from the reference and the minimization of the uncertainty of future states. Time-varying state constraints as well as time-invariant input constraints are considered, and the controllers are synthesized by semi-definite programming.

On Consensusability of Linear Interconnected Multiagent Systems and Simultaneous Stabilization

Consensusability of multiagent systems (MASs) certifies the existence of a distributed controller capable of driving the states of each subsystem to a consensus value. We study the consensusability of linear interconnected MASs (LIMASs), where, as in several real-world applications, subsystems are physically coupled. We show that consensusability is related to the simultaneous stabilizability of multiple LTI systems, and present a novel sufficient condition in form of a linear program for verifying this property. We also derive several necessary and sufficient consensusability conditions for LIMASs in terms of parameters of the subsystem matrices and the eigenvalues of the physical and communication graph Laplacians. The results show that weak physical couplings among subsystems and densely connected physical and communication graphs are favorable for consensusability. Finally, we validate our results through simulations of networks of supercapacitors and dc microgrids.

Time-Varying Optimization of LTI Systems Via Projected Primal-Dual Gradient Flows

This paper investigates the problem of regulating in real time a linear dynamical system to the solution trajectory of a time-varying constrained convex optimization problem. The proposed feedback controller is based on an adaptation of the saddle-flow dynamics, modified to take into account projections on constraint sets and output-feedback from the plant. We derive sufficient conditions on the tunable parameters of the controller (inherently related to the time-scale separation between plant and controller dynamics) to guarantee exponential and input-to-state stability of the closed-loop system. The analysis is tailored to the case of time-varying strongly convex cost functions and polytopic output constraints. The theoretical results are further validated in a ramp metering control problem in a network of traffic highways.

Online optimization of LTI systems under persistent attacks: Stability, tracking, and robustness

We study the stability properties of a closed-loop system composed of a dynamical plant and a feedback controller, the latter generating control signals that can be compromised by a malicious attacker. We consider two classes of feedback controllers: a static output-feedback controller, and a dynamical gradient-flow controller that seeks to steer the output of the plant towards the solution of a convex optimization problem. In both cases, we analyze the stability properties of the closed-loop system under a class of switching attacks that persistently modify the control inputs generated by the controllers. Our stability analysis leverages the framework of hybrid dynamical systems, Lyapunov-based arguments for switching systems with unstable modes, and singular perturbation theory. Our results reveal that, under a suitable time-scale separation between plant and controllers, the stability of the interconnected system can be preserved when the attack occurs with “sufficiently low frequency” in any bounded time interval. We present simulation results in a power-grid example that corroborate the technical findings.

Measures for the Degree of Time Variance: Application to Filter Banks

Quantifying a system’s time-invariance deviation may reveal how it represents and processes the signals. The main objective of this paper is to quantify the departure of a time-varying system from being time-invariant. In particular, the paper quantifies the time variance of filter banks using kernel representation. We define the ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> measure for the degree of time variance (MDTV <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ₂</sup> ) based on the discrete-time domain form of the kernel representation and the ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> measure (MDTV <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ_∞</sup> ) based on the matrix representation of the LPTV kernel. We show that off-diagonal matrix elements can be used to deduce the time-varying behavior of an LPTV system, proving that time-invariant and time-varying components can be separated. The matrix form of the LTI system, which is closest to an LPTV system, has been derived. This research offers a new kernel-based definition for time variance for any time-varying system and provides explicit expressions for MDTV for three distinct scenarios, namely, a single channel (SC), a uniform filter bank (UFB), and a non-uniform filter bank (NUFB). For each scenario, we determine the LPTV kernel and impulse response of the nearest LTI system to plot the MDTV versus decimation factor. Also, MDTV generated by an SC and a UFB with identical low pass filters in each channel is identical. We also observe that two-channel perfect reconstruction (PR) filter banks result in zero MDTV. Finally, we have compared the proposed approach with earlier work and demonstrated its better efficiency in terms of computational complexity.

A numerical algorithm for solving power-exponent type nonlinear inequalities with applications in calculating stabilizing parameters for LTI systems

The paper proposes a numerical algorithm for finding a solution set of a class of power-exponent type nonlinear inequalities. The proposed numerical algorithm applies specifically to the case when the power-exponent is positive irrational number, which can’t be effectively solved by using the existing methods. Firstly, by introducing a novel concept of margin variable and using the monotonicity of power-exponent function, this paper converts the problem of solving the nonlinear inequalities into a constrained nonlinear optimization (CNO) problem. Then the obtained CNO problem is solved by using the MATLAB function fmincon. Finally, a numerical algorithm to solve this class of inequalities is obtained. The proposed algorithm can find a solution set of the considered inequalities, not just one or few solutions. Further, the derived algorithm can be easily used to calculate the stabilizing parameter ranges for LTI systems based on the Routh-Hurwitz criterion. Moreover, an example of calculating the stabilizing parameters of a LTI system and an example of solving power-exponent type nonlinear inequalities are provided to show the effectiveness of the proposed algorithm.

Finite Sample Identification of Low-Order LTI Systems via Nuclear Norm Regularization

This paper studies the problem of identifying low-order linear time-invariant systems via Hankel nuclear norm (HNN) regularization. This regularization encourages the Hankel matrix to be low-rank, which corresponds to the dynamical system being of low order. We provide novel statistical analysis for this regularization, and contrast it with the unregularized ordinary least-squares (OLS) estimator. Our analysis leads to new finite-sample error bounds on estimating the impulse response and the Hankel matrix associated with the linear system using HNN regularization. We design a suitable input excitation, and show that we can recover the system using a number of observations that scales optimally with the true system order and achieves strong statistical estimation rates. Complementing these, we also demonstrate that the input design indeed matters by proving that intuitive choices, such as i.i.d. Gaussian input, lead to sub-optimal sample complexity. To better understand the benefits of regularization, we also revisit the OLS estimator. Besides refining existing bounds, we experimentally identify when HNN regularization improves over OLS: (1) For loworder systems with slow impulse-response decay, OLS method performs poorly in terms of sample complexity, (2) the Hankel matrix returned by regularization has a more clear singular value gap that makes determining the system order easier, (3) HNN regularization is less sensitive to hyperparameter choice. To choose the regularization parameter, we also outline a simple joint train-validation procedure

Data-Driven Unknown-Input Observers and State Estimation

Unknown-input observers (UIOs) allow for estimation of the states of an LTI system without knowledge of all inputs. In this letter, we provide a novel data-driven UIO based on behavioral system theory and the result known as Fundamental Lemma proposed by Jan Willems and coworkers. We give necessary and sufficient conditions on the data collected from the system for the existence of a UIO providing asymptotically converging state estimates, and propose a purely data-driven algorithm for their computation. Even though we focus on UIOs, our results also apply to the standard case of completely known inputs. As an example, we apply the proposed method to distributed state estimation in DC microgrids.

On Discrete-Time Output Negative Imaginary Systems

This letter introduces the notion of linear Discrete-time Output Negative Imaginary (D-ONI) systems. The D-ONI class is defined in the z-domain and it includes the systems having poles on the unit circle. The proposed definition involves a real parameter δ ≥ 0, which indicates the strictness properties. δ > 0 specifies the strict subset, Discrete-time Output Strictly Negative Imaginary (D-OSNI), within the stable D-ONI class. Interestingly, the new D-ONI class captures the existing D-NI systems while restricted to discrete-time LTI systems having a real, rational and proper transfer function. However, the D-OSNI systems are not identical to the existing strictly D-NI (D-SNI) subset. Instead, these two subsets intersect each other. An LMI-based state-space characterisation is derived to check the strict/non-strict D-ONI properties of a given system relying on the value of δ. The paper also establishes the connections between the discrete-time Passive and discrete-time NI systems. Finally, a closed-loop stability result is proposed for a positive feedback interconnection of two D-ONI systems without poles at z = -1 and z = +1.

Decomposition of Additive LMIs with Applications in Distributed Analysis of Interconnected LTI Systems

In this paper we present a simple technique which can be systematically used to obtain non-conservative decomposition for a class of linear matrix inequalities (LMIs) with an additive structure. By non-conservative decomposition we mean a suitable replacement of an additive LMI with a set of equivalent inequalities which are coupled by common variables. The results are applied on several stability/dissipativity analysis problems to produce analysis LMIs suitable for distributed computation.

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.

About switched affine system interconnected with fast LTI dynamics

We study an interconnection between a slow switched affine system and a fast LTI system. The lower dimensional slow and fast subsystems are computed via the singular perturbation approach. We consider the classical stabilization method based on the existence of a stable convex combination (while ignoring the fast dynamics) and we provide an LMI condition for checking ultimate boundedness when the fast dynamics is taken into account. Numerical example illustrates the main result.

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.

A New Adaptive Disturbance/Uncertainty Estimator Based Control Scheme For LTI Systems

This paper introduces a new adaptive disturbance/uncertainty estimator based control scheme for LTI systems. The aim of the proposed method is to update the nominal model directly used by the conventional disturbance observer based control architecture and approximate it to the perturbed/uncertain system using machine learning approaches. This enhances the disturbance rejection performance of the system remarkably. The performance deterioration capacity of lumped disturbances, which are the mixed effect of disturbances entering through the control channels and modeling uncertainties, are decomposed in our approach and handled separately. For this study, harmonic disturbance model and constant unstructured uncertainty model are considered, and ϵ-Support Vector Regression approach is used together with an online adaptation algorithm. A numerical example is given to demonstrate the merits and effectiveness of the proposed approach. Simulation results show that the proposed method outperforms the conventional disturbance/ uncertainty estimator based control architecture by increasing disturbance estimation performance of the system.

Positive output controllability of linear discrete–time invariant systems

Abstract This paper studies the output controllability of discrete linear time invariant systems (LTI) with non-negative input constraints. Some geometrical arguments and positive invariance concepts are used to derive the necessary and/or sufficient conditions for the positive output controllability of discrete LTI systems. The paper also provides several academic examples, which support the theoretical results.