Linear Time-invariant Control Systems Research Articles

Stability analysis of hybrid integrator-gain systems: A frequency-domain approach

The hybrid integrator-gain system (HIGS) has been introduced recently with the aim to overcome fundamental limitations of linear time-invariant (LTI) control systems. To support the analysis and design of HIGS-based controllers, in this paper a novel frequency-domain condition for stability analysis of the feedback interconnection of an LTI system and HIGS is presented. Compared to existing frequency-domain stability conditions such as the one extending the circle-criterion, the condition presented in this paper exploits explicit knowledge regarding HIGS’ switching strategy, thereby potentially providing a significantly less conservative condition. In particular, the novel condition in this paper guarantees the existence of a quadratic Lyapunov function that does not need to be positive definite within the full state space. The proposed condition can be verified graphically in a manner that is reminiscent of the classical Popov plot, as will be illustrated in an experimental case-study.

Controllability and Observability of Linear Time-Invariant Control System on Superspace

Controllability and Observability of Linear Time-Invariant Control System on Superspace

Parameter-Optimal-Gain-Arguable Iterative Learning Control for Linear Time-Invariant Systems with Quantized Error

In this paper, a parameter optimal gain-arguable iterative learning control algorithm is proposed for a class of linear discrete-time systems with quantized error. Based on the lifting model description for ILC systems, the iteration time-variable derivative learning gain in the algorithm is optimized by resolving a minimization problem regarding the tracking error energy and the learning effort amplified by a weighting factor. Further, the tracking error can be monotonically convergent to zero when the condition is guaranteed and the rate of convergence can be adjusted by scaling the weighting factor of an optimization problem. This algorithm is more innovative when compared with the existing iterative learning control algorithm for quantization systems. The innovations of this algorithm are as follows: (i) this optimization-based strategy for selecting learning gains can improve the active learning ability of the control mechanism and avoid the passivity of existing selective learning gains; (ii) the algorithm of POGAILC with data quantization can improve the convergence performance of tracking errors and reduce the negative effects of data quantization on the control performance of the logarithmic quantizer; and (iii) we provide a rigorous algorithm convergence analysis by deriving the existence of the unique solution for the optimal learning-gain vector under a singular and nonsingular tracking-error diagonalized matrix. Finally, numerical simulations are used to demonstrate the effectiveness of the algorithm.

Consistent discretization of homogeneous finite/fixed-time controllers for LTI systems

The paper proposes a discretization (sampled-time implementation) algorithm for a class of homogeneous controllers for linear time-invariant (LTI) systems preserving the finite-time and nearly fixed-time stability properties. The sampling period is assumed to be constant. Both single-input and multiple-input cases are considered. The robustness (Input-to-State Stability) of the obtained sampled-time control system is studied as well. Theoretical results are supported by numerical simulations.

A combined method for stability analysis of linear time invariant control systems based on Hermite‐Fujiwara matrix and Cholesky decomposition

AbstractIn this paper, we have developed an integrative method for checking the stability of linear time‐invariant (LTI) systems as well as nonlinear continuous‐time ones. In our method, we first apply the iterative Faddeev–Leverrier algorithm to obtain the characteristic polynomial of the LTI system. Subsequently, the associated Hermite‐Fujiwara matrix will be evaluated by a particularly efficient technique for the calculation of the Bézoutian matrices. The positive‐definiteness of the Hermite‐Fujiwara form, as the stability criterion, is next tested by performing the Cholesky decomposition. Our method is extended to assess the local stability of nonlinear continuous‐time systems with the help of the Jacobian matrix concept. The proposed method is demonstrated to approximately be 2.2 times faster than the classical Hurwitz algorithm in average, at least for matrices with dimensions less than 40, according to a performed central processing unit (CPU) time analysis. For the sake of illustration, four numerical examples are given, including dynamical models for a real‐world hydrolysis reactor and a well‐mixed bioreactor.

Topological equivalence of linear time-varying control systems

Abstract In this paper, we mainly studied the topological equivalence of linear time-varying (LTV) control system $\dot{x}\left ( t\right )=A(t)x(t)+B(t)u(t)$ defined on an interval $I \subset \mathbb{R}^{+}$. After giving a new definition of the topological equivalence, we investigated the local equivalence of LTV control systems under two new hypotheses. These hypotheses were made by the local behavior of Krylov indices (which turned out to be controllability indices for the linear time-invariant (LTI) control systems). It was found out that Krylov indices play an important role in the classification problem of LTV control systems. Compared with our former work on the topological equivalence of LTI control systems, new methods and techniques were taken to deal with new difficulties occurred for LTV control systems.

Bridging Direct and Indirect Data-Driven Control Formulations via Regularizations and Relaxations

In this article, we discuss connections between sequential system identification and control for linear time-invariant systems, often termed indirect data-driven control, as well as a contemporary direct data-driven control approach seeking an optimal decision compatible with recorded data assembled in a Hankel matrix and robustified through suitable regularizations. We formulate these two problems in the language of behavioral systems theory and parametric mathematical programs, and we bridge them through a multicriteria formulation trading off system identification and control objectives. We illustrate our results with two methods from subspace identification and control: namely, subspace predictive control and low-rank approximation, which constrain trajectories to be consistent with a nonparametric predictor derived from (respectively, the column span of) a data Hankel matrix. In both cases, we conclude that direct and regularized data-driven control can be derived as convex relaxation of the indirect approach, and the regularizations account for an implicit identification step. Our analysis further reveals a novel regularizer and a plausible hypothesis explaining the remarkable empirical performance of direct methods on nonlinear systems.

Pole Placement for MIMO LTI Systems via extended Ackermann and Greville Formulae

The Ackermann's formula of pole placement for controllable linear time invariant (LTI) systems is extended to multi input LTI systems by employing generalized inversion of the system's controllability matrix instead of square inversion in the procedure of deriving the formula. The nullspace of the controllability matrix is affinely and redundantly parameterized in the modified Ackermann's formula by means of the Greville formula for solutions of consistent over-determined linear systems of algebraic equations. The redundant nullspace parameterizing variables are constrained such that the closed loop system matrix satisfies its characteristic equation. The infinite number of solutions for the multi input gain matrix are obtained explicitly in terms of the constrained nullspace parameterizing variables. Two examples are provided to illustrate the procedure.

Zonotopic Under-Approximations of Input Reachable Sets for Controllable Linear Systems

Recent years have seen a surging interest in developing under-approximations of reachable sets due to their potential applications in control synthesis and verification. In this paper, we propose a method that yields under-approximations of finite-time input reachable sets for continuous-time controllable linear time-invariant systems with zonotopic input sets, utilizing approximations of the matrix exponential and its integral. The proposed method generates zonotopic under-approximations that converge in the sense of the Hausdorff distance. In addition, we introduce a variant of the proposed method that is better suited for applications with large time horizons. To illustrate its performance, we implement our proposed method in two numerical examples.

Stealthy attacks and attack-resilient interval observers

Industrial control systems have been frequent targets of cyber attacks during the last decade. Adversaries can hinder the safe operation of these systems by tampering with their sensors and actuators, while ensuring that the monitoring systems are not able to detect such attacks in time. This paper presents methods to design and overcome stealthy attacks on linear time-invariant control systems that estimate their state using an interval observer, in the presence of unknown but bounded noise and perturbations. We analyze scenarios in which a malicious agent compromises the sensors and/or the actuators of the system with additive attack signals to steer the state estimate outside of the bounds provided by the interval observer. We first show that maximally disruptive attack sequences that remain undetected by a linear monitor can be computed recursively via linear programming. We then design an attack-resilient interval observer for the system’s state, identifying sufficient conditions on the sensor data for such an observer to be realizable. We propose a computational method to determine optimal observer gains using semi-definite programming and compute bounds for the unknown attack signal as well. In numerical simulations, we illustrate and compare the ability of such interval observers to still provide accurate estimates when under attack.

Distinguishability criteria of conformable hybrid linear systems

Abstract We relate this article to the emerging idea of distinguishability of conformable linear hybrid time-invariant control systems. To obtain the necessary and sufficient conditions of α \alpha -distinguishability for fractional cases, we develop the Leibnitz rule for conformable derivatives. Furthermore, with the help of a study of Laplace techniques, a more simple criterion of α \alpha -distinguishability for the fractional linear system is developed.

Predictor-Feedback Prescribed-Time Stabilization of LTI Systems With Input Delay

This article first deals with the problem of prescribed-time stability of linear systems without delay. The analysis and design involve the <i>Bell polynomials</i>, <i>the generalized Laguerre polynomials</i>, <i>the Lah numbers</i>, and a suitable <i>polynomial-based Vandermonde matrix</i>. The results can be used to design a new controller&#x2014;with time-varying gains&#x2014;ensuring prescribed-time stabilization of controllable linear time-invariant (LTI) systems. The approach leads to similar results compared to Holloway <i>et al.</i> 2019, but offers an alternative and compact control design (especially for the choice of the time-varying gains). Based on the preliminary results for the delay-free case, we then study controllable LTI systems with single input and subject to a constant input delay. We design a predictor feedback with time-varying gains. To achieve this, we model the input delay as a transport partial differential equation (PDE) and build on the cascade PDE&#x2013;ordinary differential equation setting (inspired by Krstic 2009) so as the design of the prescribed-time predictor feedback is carried out based on the backstepping approach, which makes use of <i>time-varying kernels</i>. We guarantee the bounded invertibility of the backstepping transformation, and we prove that the closed-loop solution converges to the equilibrium in a prescribed time. A simulation example illustrates the results.

Calculation of robustly relatively stabilizing PID controllers for linear time-invariant systems with unstructured uncertainty

This article deals with the calculation of all robustly relatively stabilizing (or robustly stabilizing as a special case) Proportional–Integral–Derivative (PID) controllers for Linear Time-Invariant (LTI) systems with unstructured uncertainty. The presented method is based on plotting the envelope that corresponds to the trios of P–I–D parameters marginally complying with given robust stability or robust relative stability condition formulated by means of the H∞ norm. Thus, this approach enables obtaining the region of robustly stabilizing or robustly relatively stabilizing controllers in a P–I–D space. The applicability of the technique is demonstrated in the illustrative examples, in which the regions of robustly stabilizing and robustly relatively stabilizing PID controllers are obtained for a controlled plant model with unstructured multiplicative uncertainty and unstructured additive uncertainty. Moreover, the method is also verified on the real laboratory model of a hot-air tunnel, for which two representative controllers from the robust relative stability region are selected and implemented.

TDS-CONTROL: a MATLAB package for the analysis and controller-design of time-delay systems*

This paper presents TDS-CONTROL, a MATLAB package for the analysis and design of controllers for linear time-invariant systems with discrete delays. The code is based on a state-space representation of such systems in terms of delay-differential algebraic equations with input, output, and state delays. As such, a broad class of (interconnected) systems can be considered, including neutral systems. The controller design algorithms are based on optimizing a certain objective function, such as the spectral abscissa, the H-infinity norm, or the pseudo-spectral abscissa, with respect to the controller parameters. For example, to design a stabilizing controller the spectral abscissa, i.e., the real part of the right-most characteristic root, is minimized. As a strictly negative spectral abscissa is a necessary and sufficient condition for stability, the presented design method is not conservative and a stabilizing controller can be computed whenever it exits. This comes however at the cost of having to solve a non-smooth, non-convex optimization problem. The class of considered controllers consists of static and dynamic output-feedback controllers. By allowing to fix certain entries in the controller matrices, it is also possible to design structured controllers such as decentralized, overlapping and PID controllers. Finally, as a wide-range of delay systems can be considered, the software package takes the sensitivity of certain quantities, such as the spectral abscissa and the H-infinity norm, with respect to infinitesimal delay perturbations explicitly into account.

Tuning LQR Controllers: A Sensitivity-Based Approach

We introduce an approach to efficiently tune LQR controllers for linear time-invariant systems to match a prescribed closed-loop behavior, such as the one given by a reference model. The proposed approach is able to efficiently tune the LQR controller, even for high dimensional systems and is superior in terms of achieved tracking performance and other criteria with respect to global optimization methods commonly used for black-box, simulation-based, automated tuning.

Observer construction for tracking control of systems with unknown parameters

This paper deals with output-feedback tracking control of linear time-invariant systems with unknown parameters. Controller parameters are updated in real time based on the estimation of unknown parameters using multiple observers.

An Application of the Z-Transform in Extracting the Density Regulation from the Periodic Output of an Age-Structured Population Model

The application of the Z-transform, a manipulation tool from the discrete signal processing (DSP) toolbox, on an ecological model was motivated by the mathematical similarities between an age-structured fish population model with a non linear density regulation and a linear time invariant (LTI) control system. Both models include a switching mechanism in regulating stock/signal throughput in accordance with a given density limitation/set value and both models can be expressed in terms of a negative feedback loop difference equations (Getz &amp; Haight,1989; °Astr¨om &amp; Murray, 2008). In the fish model, the switching mechanism is a density regulated stock-recruitment (SR) function which models the strategies implemented by the population in keeping the vulnerable egg-larvaejuvenile densities within an environmental limitation thereof (Subbey et al, 2014). A switching mechanism is also present in control engineering, for example, in the mechanism associated with cruise control in cars which keeps traveling speed close to a chosen set value midst varying weather and road conditions (Antsaklis and Gao, 2005). In both cases, the choosing of the control action and the tuning of its parameters requires careful consideration to avoid failures such as incorrectly timed switching actions in a control plant (see Kuphaldt (2019)) and errors in estimating total allowable catch (TAC) in the fishing industry (see Borlestean et al (2015), Skagen et al (2013) and Taboadai and R. Anadn (2016)). The Z-transform has proven itself useful in tuning LTI controlmodels for a desired control action (see Orfanidis, (2010) and Smith, (1999)) and it is on this account that its application was extended to the ecological model in pursuit of a more efficient way of estimating SR parameters to simulate an already existing output. It was however found that it could not be used for parameter tuning but rather for the extraction of the SR component hidden in the output together with components resulting from the age structure itself. Such an extraction can greatly assist in the mathematical identification of the SR, reducing the complexity of its choosing as there are many different types used in the fishing industry such as the classic Beverton-Holt model, the Ricker model and Shepherd model (Myers, 2001; Iles, 1994; Shepherd, 1982). It can also be used to monitor changes in the SR over time which can indicate the presence of strategy evolution (Apaloo et al, 2009; Br¨annstr¨om et al, 2013). In 1998 Schoombie and Getz investigated the latter by subjecting the Shepherd SR to strategy optimization with regards to a parameter associated with population interventions in regulating recruitment throughput and it is because of this versatility that the Shepherd SR is chosen for the intended extraction. In true control style, Simulink, a graphic environment for designing control simulations, is used to visualize the production of the output as well as the extraction of the SR from it. This paper showcases the versatility of the Z transform and the possibilities and unexpected finds when applied to similar systems designed to regulate signals or, in this case, recruitment densities.

Decentralized state feedback control of linear time-invariant system with free-interface substructures

The state space model and local state controller of a conventional decentralized control system are based on the physical parameters or vibration modes of the whole structure. The interaction controllers between the isolated subsystems were usually ignored or they were formulated with the diagonal dominant theory. A new decentralized control method is proposed in this paper to address these disadvantages based on the free-interface substructures method. The stability of each isolated subsystem is ensured with the state space model, and the local state controllers of the open-loop subsystems are formulated from the substructural vibration modes. The interaction controller is derived from the interface forces between the coupling substructures. This ensures the stability of the close-loop whole system with no reference to the popular diagonal dominant theory. Numerical application to a six-storey plane frame under seismic excitation demonstrates that both the substructures and the whole structural system can have better control effects and stability compared to the conventional decentralized control method with more uniform and less demand on the performances of distributed dampers in the isolated subsystems.

Decentralized time-delay controller design for systems described by delay differential-algebraic equations

This paper presents a complete approach for designing stabilizing linear time-invariant decentralized finite-dimensional or retarded time-delay output feedback controllers for linear time-invariant systems of delay differential-algebraic equations. The proposed approach is based on the sequential design of the local controllers by using a centralized controller design algorithm. In this sequential design approach, the local controller to be designed at each step is determined depending on the mobility of the rightmost modes with respect to the controllers that have not yet been designed and closed with the system. Since no predefined sequence is followed, a sequence that can target the least effort and dimension for each agent can be aimed. Also, in the proposed approach, a base controller effort can be targeted for each control agent, so that the effort required to stabilize the system can be distributed among the local controllers. In the centralized controller design algorithm used for the design of each local controller, the parameters of the controllers are changed stepwise in a quasi-continuous way to shift the targeted rightmost modes towards the stable area. For a time-delay controller, the desired mode placement can be achieved by applying small changes stepwise to the elements of the matrices and the time-delays of the controller while time-delays remain always non-negative. The effect of small perturbations on the time-delays in the open-loop system or to be added by the controller to be designed is taken into account to ensure some degree of robustness against all possible perturbations on the delays. The effectiveness of the proposed design approach is demonstrated by a numerical example.

Frequency-domain stability conditions for asynchronously sampled decentralized LTI systems

This paper deals with the exponential stability analysis of decentralized, sampled-data, Linear Time Invariant (LTI) control systems with asynchronous sensors and actuators. We consider the case where each controller in the decentralized setting has its own sampling and actuation frequency, which translates to asynchrony between sensors and actuators. Additionally, asynchrony may be induced by delays between the sampling instants and actuation update instants as relevant in a networked context. The decentralized, asynchronous LTI system is represented as the feedback interconnection of a continuous-time LTI system operator and an operator that captures the effects of asynchrony induced by sampling and delay. By characterizing the properties of the operators using small-gain type Integral Quadratic Constraints (IQC), we provide criteria for exponential stability of the asynchronous, decentralized LTI state-space models. The approach provided in this paper considers two scenarios, namely the ‘large-delay’ case and the ‘small-delay’ case where the delays are larger and smaller than the sampling interval, respectively. The effectiveness of the proposed results is corroborated by a numerical example.