Linear Time-invariant Subsystems Research Articles

Multidirectional cascade of square MIMO LTI subsystems: Transfer function representation

This paper considers continuous transfer function representation for a cascade composed of N mutually interconnected square multiple-input multiple-output (MIMO) linear time-invariant (LTI) subsystems with general multidirectional input–output configurations. Such cascaded structure can arise, e.g., from the spatial discretization of one-dimensional distributed parameter systems (DPSs) described by partial differential equations (PDEs) with boundary conditions representing different configurations of boundary input signals. As shown in the paper, the transfer function matrix of the resulting cascade can be calculated using the Redheffer star product of the subsystems’ transfer function matrices, which simplifies into the usual matrix product for the unidirectional cascade. For the particular case of 2 × 2 cascade composed of rational transfer function subsystems, the recursive formulas for its boundary and distributed transfer functions have been derived for both uni- and multidirectional configurations. The well-posedness and the stability criteria are also analyzed, proving to be more restrictive for the multidirectional cascade than for the unidirectional one due to the internal positive feedback in the first case. As shown in the attached example, the presented results can be used to approximate one-dimensional distributed parameter systems using rational transfer function subsystems representing the spatial sections of the original DPS.

Finite-time stabilization of discrete-time conewise linear systems

The finite-time stabilizing control design problem for discrete-time conewise linear systems is tackled in this paper. Such a class of systems consists of the union of ordinary linear time-invariant subsystems, whose dynamics are defined in prescribed conical regions, constituting a conical partition of the state space. By imposing some cone-copositivity properties to a suitable piecewise quadratic function, two sufficient conditions are preliminarily derived concerning the system’s finite-time stability. By building on them, novel results are then presented for the system’s finite-time stabilization through a piecewise linear output feedback controller. Such results are based on the solution of feasibility problems involving sets of Linear Matrix Inequalities (LMIs). A numerical example illustrates the effectiveness of the proposed approach.

Distributed Filtering With Dynamic Event-Triggered Communication for Networked Periodic Systems

This article deals with the problem of distributed filtering of periodic systems over sensor networks under event-triggered communications, based on a piecewise impulsive method. The continuous-time periodic system is approximated by several linear time-invariant subsystems through a piecewise constant method. The sensor network deployed to the periodic system is taken into account for the actual needs of monitoring or estimation. A dynamic event-triggering mechanism is designed to reduce the unnecessary data transmission over the wireless network from the sensor to the filter as well as between each filter. Distributed peak-to-peak filters with periodic time-varying parameters are designed under the event-triggering mechanism to estimate the networked continuous-time periodic systems with the bounded and local differentiable disturbance. Then, a periodic piecewise filtering error system is modeled as an impulsive periodic piecewise system, for which sufficient conditions of a guaranteed peak-to-peak performance are proposed. The gains of the filters are obtained by the method of convex linearization. The validity of the proposed results is demonstrated by a pendulum tracking system.

Energy-Based Analysis of String Stability in Vehicle Platoons

String stability is a vital property of vehicle platoons which ensures that the system states are not amplified downstream along the string of vehicles. In this paper, a passivity-based approach to stability analysis of vehicle platoons is presented. This is motivated by a case study of an adaptive cruise-controlled platoon. We link the string stability of the platoon to the output strict passivity of each subsystem (vehicle) in the string. We show that a sufficient condition for the platoon to be string stable is that the output strict passivity index of each vehicle in the string be lower bounded by one. For a linear time-invariant (LTI) system, this means the Nyquist plot of the transfer function of each subsystem is within the unit circle. Modeling the platoon as a cascade of LTI subsystems, we present a visual tool to assess string stability. By constraining the output strict passivity index of each subsystem in a string to unity through a two-stage control strategy, the condition for string stability is met. The proposed string stability control strategy, accounting for uncertainties and delays in the drive and sensory systems, is validated using numerical simulations.

Detection of subcritical Hopf and fold bifurcations in an aeroelastic system via the Describing Function method

The paper deals with Limit Cycle Oscillations (LCOs) in a Pitch & Plunge model of the wing dynamics, where the stiffness on the plunge displacement is assumed to be an odd fifth-order polynomial. First, it is shown that the model dynamics can be equivalently described via a Lur’e system, i.e., the feedback interconnection between a linear time-invariant subsystem, whose transfer function depends on the wind speed, and a nonlinear memoryless one. Then, the Describing Function (DF) method is used to establish the existence of Predicted Limit Cycles (PLCs) whose amplitude can be analytically expressed in terms of the wind speed, once the intersections of the Nyquist diagram of the linear subsystem with the real axis are obtained. The related bifurcation diagram permits to predict that a stable equilibrium point and two LCOs (one unstable and one stable) coexist for wind speeds smaller than the flutter velocity, at which a subcritical Hopf bifurcation occurs, and larger than the value at which a cyclic fold bifurcation is detected. The closeness of each PLC to the related LCO is measured via the so-called distortion index, which also admits a direct expression in terms of the wind speed. The comparison of the predicted bifurcation diagram with that obtained via numerical simulations confirms the quite good accuracy of the PLCs obtained via the DF method, especially when the distortion index is small.

Robustness analysis and distributed control of a networked system with time-varying delays

This paper is concerned with the robustness analysis and distributed output feedback control of a networked system with uncertain time-varying communication delays. This system consists of a collection of linear time-invariant subsystems that are spatially interconnected via an arbitrary directed network. Using a dissipation inequality that incorporates dynamic hard IQCs (integral quadratic constraints) for the delay uncertainties, we derive some sufficient robustness conditions in the form of coupled linear matrix inequalities, in which the coupled parts reflect the interconnection structure of the system. We then provide a procedure to construct a distributed controller to ensure the robust stability of the closed-loop system and to achieve a prescribed l2-gain performance. The effectiveness of the proposed approach is demonstrated by some numerical examples.

Joint Cross-Layer Optimization in Real-Time Networked Control Systems

Networked control system (NCS) refers to a set of control loops that are closed over a communication network. In this article, the joint operation of control and networking for NCS is investigated wherein the network serves the sensor-to-controller communication links for multiple stochastic linear time-invariant (LTI) subsystems. The sensors sample packets based on the observed plant state, which they send over a shared multihop network. The network has limited communication resources, which need to be assigned to competing links to support proper control loop operation. In this setup, we formulate an optimization problem to minimize the weighted-sum linear-quadratic-Gaussian cost of all loops, taking into account the admissible sampling, control, congestion control (CC), and scheduling policies. Under some mild assumptions on the sampling frequencies of the control loops and the communication network, we find the joint optimal solution to be given by a certainty equivalence control with a threshold-based sampling policy, as well as a back-pressure-type scheduler with a simple pass-through CC. The interface between network and control loops is identified to be the buffer state of the sensor node, which can be interpreted as network price for sampling a packet from the control perspective. We validate our theoretical claims by simulating NCSs comprised of multiple LTI stochastic control loops communicating over a two-hop cellular network.

Finite‐time control of periodic systems with event‐triggering mechanisms

This study is concerned with the event-triggered finite-time control of periodic systems via a piecewise method. The continuous-time periodic system is formulated by finite-number linear time-invariant subsystems, of which the system parameters are determined by the piecewise approximation method. In consideration of the mitigation of network communication in networked control loops, some event-triggering mechanisms are designed in terms of the networks between the sensor (or controller) and the controller (or actuator). Three different event-triggered control schemes are designed for the networked periodic system. The finite-time stability of the resulting closed-loop system is analysed by using a Lyapunov function approach with a well-defined matrix-valued function. The controller gains depending on the piecewise parameters are thus designed with several matrix inequalities that can be solved by off-line procedures. Finally, a numerical example is provided to illustrate the effectiveness of the proposed design scheme.

A Novel Fuzzy PI Control Approach for Nonlinear Processes

Classical fuzzy logic controllers fail to control nonlinear systems, which have strong nonlinearity around a certain operating point. In this case, if it is possible, the block-oriented modeling approach could be adopted to separate the highly nonlinear system into two subsystems as a linear time-invariant subsystem and a memoryless nonlinear subsystem. Thereby, a control operation could be carried out on the linear time-invariant block to control the whole system. Respecting this approach, here, a highly nonlinear system (neutralization process) was modeled by using the ARX–polynomial cascade connection. By this means, an opportunity is obtained that the inverse polynomial is utilized to be able to control the ARX subsystem by a classical fuzzy PI controller. And therefore, the fuzzy PI controller performance in controlling nonlinear processes was enhanced. The success of the new approach was confirmed by the results obtained from the control operation carried out in different pH regions for a chemical neutralization process. And also, the results show that the proposed control strategy is as successful as the multiregion fuzzy PI controller in the highly nonlinear process control.

Dissipative observers for discrete-time nonlinear systems

This paper addresses the problem of designing a state observer for a class of nonlinear discrete-time systems using the dissipativity theory. We show that the dissipative observation methodology, originally proposed by one of the authors for continuous-time nonlinear systems, can be extended to the discrete-time case. For constructing a convergent observer, the methodology is applied to the nonlinear estimation error dynamics, which is decomposed into a discrete-time Linear Time-Invariant (LTI) subsystem in the forward loop, connected to a time-varying static nonlinearity in the feedback loop. In order to assure asymptotic stability of the closed-loop, complementary dissipativity conditions are imposed on each of the subsystems: (i) the static nonlinearity is required to be dissipative with respect to a quadratic supply rate, and (ii) the observer gains are designed such that the LTI system is dissipative with respect to a complementary supply rate. As in the continuous time framework, the proposed method includes as special cases, unifies and generalizes some observer design methods proposed previously in the literature. A great advantage of the Dissipative Observer Design Method proposed here is that it leads to Matrix Inequalities for the design of the observer gains, and these can be usually converted into Linear Matrix Inequalities (LMI’s). The results are illustrated using Chua’s Chaotic system.

Distributed Kalman filter in a network of linear systems

This paper is concerned with the problem of distributed Kalman filtering in a network of interconnected subsystems with distributed control protocols. We consider networks, which can be either homogeneous or heterogeneous, of linear time-invariant subsystems, given in the state-space form. We propose a distributed Kalman filtering scheme for this setup. The proposed method provides, at each node, an estimation of the state parameter, only based on locally available measurements and those from the neighbor nodes. The special feature of this method is that it exploits the particular structure of the considered network to obtain an estimate using only one prediction/update step at each time step. We show that the estimate produced by the proposed method asymptotically approaches that of the centralized Kalman filter, i.e., the optimal one with global knowledge of all network parameters, and we are able to bound the convergence rate. Moreover, if the initial states of all subsystems are mutually uncorrelated, the estimates of these two schemes are identical at each time step.

VSC-HVDC system robust stability analysis based on a modified mixed small gain and passivity theorem

In a VSC-HVDC system, when the input power to the converter is controlled to a positive constant, the converter acts a differential negative resistance — Po/i20, which may cause instability in case of a connected weak AC-grid. In order to analyze the robust stability of such a system, a modified mixed small gain and passivity theorem is proposed, which shows that the negative feedback interconnection of two linear time invariant stable subsystems (converter admittance and AC-grid impedance) is input to output stable if in certain frequency bands, both subsystems are passive and at least one of them is strictly passive; at the remaining frequency bands, the loop gain is to be strictly less than one. Under different AC-grid topologies, the stability sufficient conditions with respect to the AC-grid inductance are studied. It shows that for AC-grid consists of series RgLg-circuit, an introduced series capacitor can improve the system robustness but not for the shunt capacitor.

Strong Structural Input and State Observability of LTV Network Systems with Multiple Unknown Inputs

This paper studies linear time-varying (LTV) network systems affected by multiple unknown inputs. The goal is to reconstruct both the initial state and the unknown input. The main result is a characterization of strong structural input and state observability, i.e., the conditions under which both the whole network state and the unknown input can be reconstructed for all system matrices that share a common zero/non-zero pattern. This characterization is in terms of strong structural observability of a suitably-defined linear time-invariant (LTI) subsystem.

Frequency-domain subsystem identification with application to modeling human control behavior

We present a frequency-domain subsystem identification algorithm that identifies unknown feedback and feedforward subsystems that are interconnected with a known subsystem. This method requires only accessible input and output measurements, applies to linear time-invariant subsystems, and uses a candidate-pool approach to ensure asymptotic stability of the identified closed-loop transfer function. We analyze the algorithm in the cases of noiseless and noisy data. The main analytic result of this paper shows that the coefficients of the identified feedback and feedforward transfer functions are arbitrarily close to the true coefficients if the data noise is sufficiently small and the candidate pool is sufficiently dense. This subsystem identification approach has application to modeling the control behavior of humans interacting with and receiving feedback from a dynamic system. We apply the algorithm to data from a human-in-the-loop experiment to model a human’s control behavior.

Decentralized string-stability analysis for heterogeneous cascades subject to load-matching requirements

A heterogeneous cascade of stable linear time-invariant subsystems is studied in terms of the spatial and temporal propagation of boundary conditions. The particular context requires constant spatial boundary conditions to be asymptotically matched by the interconnection signals along the string (e.g., to match supply to demand in steady state). Furthermore, the transient response associated with a step change in the spatial boundary condition must remain bounded across space in a string-length independent fashion. With this in mind, an infinite cascade abstraction is considered. A corresponding decentralized string-stability certificate for the desired behaviour is established in terms of the subsystem $$H_\infty $$H? norms, via Lyapunov-type analysis of a two-dimensional model in Roesser form. Verification of the certificate implies uniformly bounded interconnection signals in response to the following system inputs: (i) a square-summable (across space) sequence of initial conditions; and (ii) a uniformly-bounded (across time) finite-energy input applied as the spatial boundary condition (e.g., finite duration on-off pulse). The decentralized nature of the certificate facilitates subsystem-by-subsystem design of local controllers that achieve string-stable behaviour overall. This application of the analysis is explored within the context of a scalable approach to the design of distributed distant-downstream controllers for the sections of an automated irrigation channel.

An immune-inspired, information-theoretic framework for blind inversion of Wiener systems

This work proposes a new approach to the blind inversion of Wiener systems. A Wiener system is composed of a linear time-invariant (LTI) sub-system followed by a memoryless nonlinear function. The goal is to recover the input signal by knowing just the output of the Wiener system, and the straightforward scheme to achieve this is called the Hammerstein system – apply a memoryless nonlinear mapping followed by a LTI sub-system to the output signal of the Wiener system. If the input of the Wiener system is originally iid and some mild conditions are satisfied, the inversion is possible. Based on this statement and the limitations of relevant previous works, a solution is proposed combining (i) immune-inspired optimization algorithms, (ii) information theory and (iii) IIR filters that yield a robust scheme with a relatively reduced risk of local convergence. Experimental results indicated a similar or superior performance of the new approach, in comparison with two other blind methodologies.

Parametric identification of parallel Wiener–Hammerstein systems

Block-oriented nonlinear models are popular in nonlinear modeling because of their advantages to be quite simple to understand and easy to use. To increase the flexibility of single branch block-oriented models, such as Hammerstein, Wiener, and Wiener–Hammerstein models, parallel block-oriented models can be considered. This paper presents a method to identify parallel Wiener–Hammerstein systems starting from input–output data only. In the first step, the best linear approximation is estimated for different input excitation levels. In the second step, the dynamics are decomposed over a number of parallel orthogonal branches. Next, the dynamics of each branch are partitioned into a linear time invariant subsystem at the input and a linear time invariant subsystem at the output. This is repeated for each branch of the model. The static nonlinear part of the model is also estimated during this step. The consistency of the proposed initialization procedure is proven. The method is validated on real-world measurements using a custom built parallel Wiener–Hammerstein test system.

Hybrid State Observer with Modulated Correction for Periodic Systems under Intrinsic Impulsive Feedback

A new observer structure for hybrid systems comprised of a linear time-invariant subsystem and immeasurable (intrinsic) impulsive feedback is suggested. It brings about significant improvements in the state estimation error settling time in comparison with that of an earlier considered static feedback observer by making use of the weighted output estimation error in the calculation of the discrete part of the observer.

Stability of distributed heterogeneous systems with static nonlinear interconnections

Abstract This paper studies the problem of L 2 stability of a system with distributed heterogeneous dynamic nodes (or subsystems) that are connected by static nonlinear links. Both the heterogeneities of the subsystems and the nonlinearities of the interconnections are considered. First, a sufficient and necessary condition is given, which shows the equivalence between the L 2 stability and an integral quadratic inequality defined by a specially structured operator. Then, under the assumption that the nonlinearities of the interconnections satisfy the sector conditions, an algebraic constraint on the interconnection is established. Based on this, a sufficient condition that the system is finite-gain L 2 stable is obtained. For the case of linear time invariant (LTI) subsystems, a frequency domain condition is given, which is numerically solvable by using the Kalman–Yakubovic–Popov (KYP) lemma. Finally, a numerical example is included to demonstrate the use of the results obtained.

Wiener Model Identification of Blast Furnace Ironmaking Process

To account for the nonlinearity of blast furnace ironmaking process, a nonlinear Wiener model identification algorithm is presented. The system consists of a linear time invariant (LTI) subsystem followed by a static nonlinearity. The inverse of the nonlinearity is assumed to be a linear combination of known nonlinear basis functions and the linear subspace algorithm is used to identify the model. The inputs to the model are parameters regarded to be most responsible for the fluctuation of thermal state in blast furnace while the output to the model is silicon content in hot metal. The identified Wiener model is then tested on datasets obtained from No. 6 Blast Furnace from Baotou Steel. It is found that the blast furnace of concern is a short memory system, so that for each prediction the Wiener method is retrained. It is shown that the retrained model well improves the predictive accuracy.