State-space Matrices Research Articles

Aircraft Parameter Estimation Considering Process and Measurement Noise

A practical formulation is proposed for aircraft parameter estimation using the filter-error method, which is a maximum likelihood estimator for dynamic systems with both process noise and measurement noise. The novelty of the proposed formulation is that by accurately estimating the measurement noise covariance matrix using a time series analysis method, the remaining unknowns (which include the unknown parameters in the state-space matrices and the process noise covariance matrix) become decorrelated and can be estimated simultaneously in a straightforward manner. The approach is demonstrated using simulation data and flight test data from a subscale airplane. Results indicate that the proposed algorithm can produce accurate modeling results when both measurement noise and process noise are present in the data.

State-Space Modeling, Design, and Analysis of the DC-DC Converters for PV Application: A Review

Small-signal models of dc-dc converters are often designed using a state-space averaging approach. This design can help discuss and derive the control-oriented and other frequency-domain attributes, such as input or output impedance parameters. This paper aims to model the dc-dc converters for PV application by employing a capacitor on the input side. The modeling, design, and analysis of the dc-dc converters regarding the input capacitor is limited in the literature. Five dc-dc converters, including buck, boost, buck-boost, ĆUK, and SEPIC converters, are designed and implemented using the state-space average modeling approach in MATLAB/Simulink. The circuit topology of each converter and the state-space matrices are derived considering every constraint. A rigorous and compelling analysis of the dc-dc converters is carried out to compare system stability and, ultimately, the dynamic performance. The output of the resulting small-signal models has been demonstrated in the time-domain against topology simulations. All the converters are exposed to unpredictable weather conditions and the simulations are carried out in the PSIM software. The perturb and observe (P&O) maximum power point tracking (MPPT) algorithm is applied in all the converters to ensure maximum power point (MPP) achievement. The results showcase that the boost converter outperforms all other converters in terms of stability, settling time, and overshoot.

On the Certainty-Equivalence Approach to Direct Data-Driven LQR Design

The linear quadratic regulator (LQR) problem is a cornerstone of automatic control, and it has been widely studied in the data-driven setting. The various data-driven approaches can be classified as indirect (i.e., based on an identified model) versus direct or as robust (i.e., taking uncertainty into account) versus certainty-equivalence. Here we show how to bridge these different formulations and propose a novel, direct, and regularized formulation. We start from indirect certainty-equivalence LQR, i.e., least-square identification of state-space matrices followed by a nominal model-based design, formalized as a bi-level program. We show how to transform this problem into a single-level, regularized, and direct data-driven control formulation, where the regularizer accounts for the least-square data fitting criterion. For this novel formulation we carry out a robustness and performance analysis in presence of noisy data. In a numerical case study we compare regularizers promoting either robustness or certainty-equivalence, and we demonstrate the remarkable performance when blending both of them.

Alternative method to include the frequency-effect on transmission line parameters via state-space representation

This article describes an alternative method and investigates its performance on the transient responses considering a frequency-dependent lumped parameter model (FDLPM) to represent the transmission line (TL) in the literature. This model includes the fundamental behavior of the frequency dependency on longitudinal parameters, and it is developed directly in the time domain (without the use of frequency-to-time conversion tools). The alternative method individually solves a system of state-space equations for each π circuit of the LPM. Consequently, this method reduces the order of the state-space matrices, providing excellent performance associated with lower computational efforts (faster speed) compared to that obtained with the classical method. The alternative method was used to calculate the electromagnetic transients (current and voltages) in the single- and three-phase transmission lines subjected to energization maneuver and lightning strike for several scenarios at the receiving end. All responses are calculated using a programming code in MATLAB®. Results demonstrated that the alternative method closely follows the responses calculated with the classical, indicating its excellent accuracy. Besides, the smaller dimension of state-space matrices lead to low computational times, varying from 230 to 300 times smaller, depending on the line configuration compared to the other method, assuming the same fixed time step.

An Explicit Dual Control Approach for Constrained Reference Tracking of Uncertain Linear Systems

A finite horizon optimal tracking problem is considered for linear dynamical systems subject to parametric uncertainties in the state-space matrices and exogenous disturbances. A suboptimal solution is proposed using a model predictive control (MPC) based explicit dual control approach which enables active uncertainty learning. A novel algorithm for the design of robustly invariant online terminal sets and terminal controllers is presented. Set membership identification is used to update the parameter uncertainty online. A predicted worst-case cost is used in the MPC optimization problem to model the dual effect of the control input. The cost-to-go is estimated using contractivity of the proposed terminal set and the remaining time horizon, so that the optimizer can estimate future benefits of exploration. The proposed dual control algorithm ensures robust constraint satisfaction and recursive feasibility, and navigates the exploration-exploitation trade-off using a robust performance metric.

State space analysis and control transfer function of a flyback converter

Flyback converters are used in automotive power supply applications for their advantage of galvanic isolation and the low number of components on a small PCB area. This paper is dedicated to state space analysis of flyback converter with parasitic elements. The Euler method also uses state-space matrices of the proposed flyback converter with nonideal flyback converter with a Bode plot of both open-loop and closed-loop systems. The advantage of this approach is to determine the stability of the open-loop system from a mathematical model and to estimate suitable PI tuning values for the closed-loop system. Transfer function and Bode plots have been obtained from state space matrices in MATLAB.

Initialization Approach for Nonlinear State-Space Identification via the Subspace Encoder Approach

The SUBNET neural network architecture has been developed to identify nonlinear state-space models from input-output data. To achieve this, it combines the rolled-out nonlinear state-space equations and a state encoder function, both parameterised as neural networks The encoder function is introduced to reconstruct the current state from past input-output data. Hence, it enables the forward simulation of the rolled-out state-space model. While this approach has shown to provide high-accuracy and consistent model estimation, its convergence can be significantly improved by efficient initialization of the training process. This paper focuses on such an initialisation of the subspace encoder approach using the Best Linear Approximation (BLA). Using the BLA provided state-space matrices and its associated reconstructability map, both the state-transition part of the network and the encoder are initialized. The performance of the improved initialisation scheme is evaluated on a Wiener-Hammerstein simulation example and a benchmark dataset. The results show that for a weakly nonlinear system, the proposed initialisation based on the linear reconstructability map results in a faster convergence and a better model quality.

An EM Algorithm for Lebesgue-sampled State-space Continuous-time System Identification*

This paper concerns the identification of continuous-time systems in state-space form that are subject to Lebesgue sampling. Contrary to equidistant (Riemann) sampling, Lebesgue sampling consists of taking measurements of a continuous-time signal whenever it crosses fixed and regularly partitioned thresholds. The knowledge of the intersample behavior of the output data is exploited in this work to derive an expectation-maximization (EM) algorithm for parameter estimation of the state-space and noise covariance matrices. For this purpose, we use the incremental discrete-time equivalent of the system, which leads to EM iterations of the continuous-time state-space matrices that can be computed by standard filtering and smoothing procedures. The effectiveness of the identification method is tested via Monte Carlo simulations.

Conversion From Unstructured LTI Controllers to Observer-Structured Ones for LPV Systems

This paper considers the conversion problem from unstructured linear time-invariant (LTI) controllers to observer-structured LTI controllers, whose structure is similar to but not exactly the same as the so-called “Luenberger observer–based controllers,” for linear parameter-varying (LPV) plant systems. In contrast to Luenberger observer–based controllers, observer-structured LTI controllers can be defined and constructed even if the plant systems are given as LPV systems. In the conversion problem, the state-space matrices of the observer-structured LTI controller are parameterized with those of the given unstructured LTI controller, one free matrix, and a state transformation matrix. We also show a method to obtain the optimal state transformation matrix with respect to the convergence of the discrepancy between the plant state and the observer-structured controller state for a stochastically defined non-zero initial plant state. Several toy examples are included to illustrate the effectiveness and the usefulness of observer-structured LTI controllers, and the utility of the proposed conversion parametrization.

Non-characteristic harmonics in modular multilevel converters—An analytical approach

This paper analyzes the harmonic content of the arm currents and voltages of the modular multilevel converter (MMC) by using the dynamic phasor (DP) theory. DP modeling of the MMC facilitates a study without relying on simplifying assumptions on the harmonic content of the currents and voltages. In doing so, the DP-MMC model used in this paper reveals generalized conditions under which non-characteristic harmonics can exist. These conditions offer a detailed insight into the transient behavior of harmonics in the MMC. An open-loop DP model of the MMC is developed, which has the harmonics of the arm currents and voltages as independent state variables. The linearized state-space matrices of the DP model are assessed under different grounding options at the ac side of the MMC, and the relationship between characteristic and non-characteristic harmonics of the MMC voltages and currents is discussed.

Extraction of Electrochemical Impedance Spectra from Physical Models

Electrochemical impedance spectroscopy (EIS) is used widely to assist the understanding and interpretation of electrochemical cells, including batteries, fuel cells, electrolyzers, etc. The measured spectra, often illustrated in terms of Nyquist plots, are typically interpreted in terms of equivalent-circuit models. The present approach is to interpret EIS using physical models. In principle, virtually any physical model that represents physics and chemistry in terms of conservation equations can be written abstractly as Eqs. 1-2, where x is the state vector, u is the actuation vector, yis the observable vector, and p is the physical param eter vector. The model can be represented in locally linear state-space form as Eqs. 3-4, where the Jacobian matrices A, B, C, and Dcan be established from local linearization as Eq. 5. The present paper compares two approaches to determine the state-space matrices. Assuming one has access to the physical model’s source code, the matrices can be evaluated by direct computational differentiation. Alternatively, the matrices can be evaluated using Pseudo Random Binary Sequence (PRBS) perturbations. With these matrices in hand, using the Laplace transform between actuation to observable, the complex impedance (i.e., real and imaginary parts of the impedance at frequency ω) can be evaluated as Eq. 6, where I is the identity matrix. By extracting the EIS from a physical model, the relationships between measurable parameters, such as cell dimensions, ion mobilities, etc., and the complex impedance can be established. These relationships are much more physically meaningful than those that can be inferred from equivalent-circuit models. The presentation includes examples of EIS derived from pseudo-two-dimensional (P2D) Li-ion battery models. Figure 1

Generation of Accessible Sets in the Dynamical Modeling of Quantum Network Systems

In this paper, we consider the dynamical modeling of a class of quantum network systems consisting of qubits. Qubit probes are employed to measure a set of selected nodes of the quantum network systems. For a variety of applications, a state space model is a useful way to model the system dynamics. To construct a state space model for a quantum network system, the major task is to find an accessible set containing all of the operators coupled to the measurement operators. This paper focuses on the generation of a proper accessible set for a given system and measurement scheme. We provide analytic results on simplifying the process of generating accessible sets for systems with a time-independent Hamiltonian. Since the order of elements in the accessible set determines the form of state space matrices, guidance is provided to effectively arrange the ordering of elements in the state vector. Defining a system state according to the accessible set, one can develop a state space model with a special pattern inherited from the system structure. As a demonstration, we specifically consider a typical 1D-chain system with several common measurements, and employ the proposed method to determine its accessible set.

Type-2 fuzzy instrumental variable algorithm for evolving neural-fuzzy modeling of nonlinear dynamic systems in noisy environment

In neural-fuzzy modeling, dealing with dynamic systems in a noisy environment is a relevant issue due to uncertainties in the experimental data. In this sense, a type-2 fuzzy instrumental variable based learning algorithm for evolving neural-fuzzy modeling is proposed in this paper. For antecedent adaptation, an Extend Kalman Filter based evolving type-2 fuzzy algorithm is adopted, where the participatory concept is used to mitigate the effects of outliers in the data flow. For consequent estimation, a type-2 fuzzy instrumental variable based subspace identification method is proposed. From the data flow, the instrumental variables are computed via recursive singular spectral analysis and used to estimate the fuzzy Markov parameters. From Markov parameters, the linear state-space matrices are obtained, recursively. Results illustrate the efficiency of the proposed methodology compared to other prominent approaches in the literature for identifying a nonlinear dynamic system with discontinuous function corrupted by colored noise and online identification of a 2DoF helicopter with correlated noise.

Probabilistic Stabilizability Certificates for a Class of Black-Box Linear Systems

We provide out-of-sample certificates on the controlled invariance property of a given set with respect to a class of black-box linear systems. Specifically, we consider linear time-invariant models whose state space matrices are known only to belong to a certain family due to a possibly inexact quantification of some parameters. By exploiting a set of realizations of those undetermined parameters, verifying the controlled invariance property of the given set amounts to a linear program, whose feasibility allows us to establish an a-posteriori probabilistic certificate on the controlled invariance property of such a set with respect to the nominal linear time-invariant dynamics. The proposed framework is applied to the control of a networked multi-agent system with unknown weighted graph.

Generic Delay-L Left Invertibility of Structured Systems

This paper studies structured systems, namely linear systems where the state-space matrices have zeros in some fixed positions and free parameters in all other entries. This paper focuses on time-invariant systems in discrete time affected by an unknown input, and their delay-L left invertibility, namely the possibility to reconstruct the input sequence from the output sequence, assuming that the initial state is known, and requiring that the inputs can be reconstructed up to L time steps before the current output. Building upon classical results on linear systems theory and on structured systems, a graphical characterization is obtained of the integers L for which a structured system is generically delay-L left invertible.

Distributed estimation and control of a four-tank process

A multivariable process of four interconnected water tanks is considered for modeling and control. The objective of the current study is to design and implement a distributed control and estimation (DEC) for a multivariable four-tank process. Distributed model and inter-nodal communication structure are derived from global state–space matrices, thus combining the topology of plant flow sheet and the interaction dynamics across the plant subunits. Using experimental data, the process dynamics and disturbance effects are modeled. A typical lab-scale system was simulated and the obtained results demonstrated the potential of the DEC algorithm.

Uncertainty quantification of input matrices and transfer function in input/output subspace system identification

The transfer function of a linear mechanical system can be defined in terms of the quadruplet of state-space matrices (A,B,C,D) that can be identified from input and output measurements with subspace-based system identification methods. The estimation of the quadruplet has been well studied in the literature from both theoretical and practical viewpoints. Nonetheless, a practical algorithm for uncertainty quantification of its estimation errors and the uncertainty of the resultant parametric transfer function is missing in the context of subspace identification. For several output-only and input/output subspace methods, the covariance related to the matrices (A,C) and to the resulting modal parameters can be effectively obtained with recently developed first-order perturbation-based schemes, while the corresponding uncertainty quantification for the input-related matrices (B,D) is missing. In this paper, explicit expressions for the covariance related to matrices (B,D) are developed, and applied to the covariance estimation of the resulting transfer function. The proposed schemes are validated on a simulated data of a mechanical system and are applied to laboratory measurements of a plate.

Application and Performance Evaluation of Model Predictive Controllers on Driving Force Based Reactive Distillation Processes

In this work, integrated design and control of Reactive Distillation (RD) column operating under Model Predictive Controller (MPC) is presented. At first, a design target is defined, and the RD column is designed based on the element concept at the maximum driving force. After that, the steady state simulation is carried out in ASPEN PLUS that satisfies the design target. Next, the flow driven dynamic system is linearized in ASPEN PLUS Dynamics and the state space matrices are obtained. Using the matrices, MPC closed-loop simulation is carried out in MATLAB. To quantify both the steady state and dynamic operation, a multi objective function is defined that takes into account both the design (total energy consumption, total carbon footprint), and the control performance (Integral Absolute Error, total variation of input, relative gain array, Niederlinski Index). For further verification, an alternative design based on non-maximal driving force is also obtained, and the corresponding steady state and dynamic simulation is performed. The closed-loop performance of MPC controller under feed disturbance shows that the process that is designed at the maximum driving force exhibit better control performance compared to the design alternative that does not take into account the maximum available driving force.
 Chemical Engineering Research Bulletin 21(2020) 49-53

Virtual Capacitor Concept for Computationally Efficient and Flexible Real-Time MMC Model

This paper demonstrates the splitting of the modular multilevel converter real-time simulation model into several independent parts through the use of the virtual capacitor concept. As a result, the number of state-space matrices the real-time solver needs to take into account gets significantly reduced, offering the possibility for substantial reduction of the simulation step size. Consequently, real-time simulation quality increases. The proposed concept was verified on a large-scale hardware-in-the-loop system comprising seven RT Boxes, where the model of the physical system is deployed, and ABB PEC800 industrial controller, where control algorithms of the real power hardware are deployed and executed.

Extension of State Space and Lyapunov Matrices

One of the main problems in the application of the theory of Lyapunov-Krasovskii functionals is the construction of corresponding Lyapunov matrices. Recently, it was noted that systems with distributed delay and exponential kernel may be rewritten by the introduction of auxiliary variables as systems with one delay. A suggestion was made that one can use the Lyapunov matrix of the latter system to obtain the Lyapunov matrix of the original system. In this article, we establish precise relationships between these two systems and their Lyapunov matrices. We show that if there exists a Lyapunov matrix of the extended system then it can be used to compute a Lyapunov matrix of the nominal system. We demonstrate that this method fails for certain systems and establish necessary and sufficient conditions for the extended system to admit a Lyapunov matrix.