Linear Time-invariant State Research Articles

Image based Modeling and Control for Batch Processes

This manuscript addresses the problem of leveraging thermal images for modeling and feedback control, specifically tailored for terminal quality control of batch processes. The primary objective, common in many batch processes, is to produce products with quality variables aligning with user specifications, available for measurement only at batch termination, precluding the direct use of classical control strategies. Furthermore, in many instances, traditional online sensors such as thermocouples may not be available, but instead spectral inputs like thermal images or acoustic data may be more readily available for feedback control. The challenge is to not only use the non-traditional sensor data for building a dynamic model but also to use that model for terminal quality control. The proposed approach involves a multi-layered modeling strategy. Initially, a dimensionality reduction technique is employed to condense the high-dimensional image into a set of representative outputs. Subsequently, subspace identification (SSID) is applied to develop a Linear Time-Invariant (LTI) State Space (SS) model between the inputs and the reduced outputs. Finally, a Partial Least Squares (PLS) model is constructed linking the terminal states of a batch (identified using SSID) with the product qualities obtained for that specific batch. This model is then incorporated into a Model Predictive Control (MPC) formulation. The effectiveness of the MPC is illustrated by showcasing its capability to generate products of high quality by deploying the MPC on a bi-axial lab-scale rotational molding setup.

Study on Efficient Sensor Node Selection for Observability Gramian Optimization⋆

This study attempts a practical comparison of optimization methods for sensor node selection to efficiently monitor large-scale dynamical systems represented by linear time-invariant state space models. Sensor measurements are evaluated based on an observability measure, the matrix determinant of the observability Gramian. This study confirms the applicability of selection strategies, namely, a convex relaxation method using semidefinite programming, a greedy maximization and its approximation that considers the gradient of the observability measure. Examples based on numerical and real-world experiments illustrate the effectiveness of the selection algorithms in terms of their optimization measures and the run time for the selection.

On the herdability of linear time-invariant systems with special topological structures

In this paper we investigate the herdability property, namely the capability of a system to be driven towards the (interior of the) positive orthant, for linear time-invariant state space models. Herdability of certain matrix pairs (A,B), where A is the adjacency matrix of a multi-agent network, and B a selection matrix that singles out a subset of the agents (the “network leaders”), is explored. The cases when the graph associated with A, G(A), is directed and clustering balanced (in particular, structurally balanced), or it has a tree topology and there is a single leader, are investigated.

Full and Partial Eigenvalue Placement for Minimum Norm Static Output Feedback Control

The controller design for linear time-invariant state space systems seems to be straightforward and well established. This is not true for static output feedback control, which is still a challenging task. This paper deals with controller design based on eigenvalue assignment. We consider the placement of distinct as well as multiple real eigenvalues or complex conjugate pairs. The desired eigenvalue configurations are characterised in terms of algebraic divisibility of the characteristic polynomial of the closed-loop system. We also consider the problem of partial eigenvalue placement, where not all eigenvalues are fixed by feedback. Degrees of freedom in the controller design are used for the minimization of various matrix norms of the feedback gain matrix.

Discretization and Model Reduction Error Estimation of Interconnected Dynamical Systems

In this paper a novel approach for estimating both discretization and model order reduction error is presented. Discretization is carried out using finite element method. Finite element models are converted to linear time-invariant state space models. Using the generalized square root balanced truncation method a series of reduced models are obtained. It is shown that for special class of passive and dissipative dynamical systems upper a-posteriori reduction error bound can be modeled less conservatively. Estimation of both discretization and model reduction error is further refined using v-gap metric. Quality of the obtained reduced order model is verified in the frequency domain. The proposed error estimation approach is demonstrated on a numerical example.

Optimal experimental design for linear time invariant state–space models

The linear time invariant state–space model representation is common to systems from several areas ranging from engineering to biochemistry. We address the problem of systematic optimal experimental design for this class of model. We consider two distinct scenarios: (i) steady-state model representations and (ii) dynamic models described by discrete-time representations. We use our approach to construct locally D-optimal designs by incorporating the calculation of the determinant of the Fisher Information Matrix and the parametric sensitivity computation in a Nonlinear Programming formulation. A global optimization solver handles the resulting numerical problem. The Fisher Information Matrix at convergence is used to determine model identifiability. We apply the methodology proposed to find approximate and exact optimal experimental designs for static and dynamic experiments for models representing a biochemical reaction network where the experimental purpose is to estimate kinetic constants.

Exponential Stabilization of Linear Time-Varying Differential Equations with Uncertain Coefficients by Linear Stationary Feedback

We consider a control system defined by a linear time-varying differential equation of n-th order with uncertain bounded coefficients. The problem of exponential stabilization of the system with an arbitrary given decay rate by linear static state or output feedback with constant gain coefficients is studied. We prove that every system is exponentially stabilizable with any pregiven decay rate by linear time-invariant static state feedback. The proof is based on the Levin’s theorem on sufficient conditions for absolute non-oscillatory stability of solutions to a linear differential equation. We obtain sufficient conditions of exponential stabilization with any pregiven decay rate for a linear differential equation with uncertain bounded coefficients by linear time-invariant static output feedback. Illustrative examples are considered.

New integer-order approximations of discrete-time non-commensurate fractional-order systems using the cross Gramian

This paper presents a new approach to modeling of linear time-invariant discrete-time non-commensurate fractional-order single-input single-output state space systems by means of the Balanced Truncation and Frequency Weighted model order reduction methods based on the cross Gramian. These reduction methods are applied to the specific rational (integer-order) FIR-based approximation to the fractional-order system, which enables to introduce simple, analytical formulae for determination of the cross Gramian of the system. This leads to significant decrease of computational burden in the reduction algorithm. As a result, a rational and relatively low-order state space approximator for the fractional-order system is obtained. A simulation experiment illustrates an efficiency of the introduced methodology in terms of high approximation accuracy and low time complexity of the proposed method.

Maximum likelihood identification of stable linear dynamical systems

This paper concerns maximum likelihood identification of linear time invariant state space models, subject to model stability constraints. We combine Expectation Maximization (EM) and Lagrangian re ...

A Stealth Integrity Targeted Cyber‐Attack in Distributed Electric Power Networks with Local Model Information

AbstractElectric power networks are critical infrastructures, and their correct operation is of vital importance. Nowadays, these systems are prone to cyber‐attacks because of new vulnerabilities in the system and access to shared networks. In this paper, a novel Stealth Integrity Targeted Attack (SITA) is proposed in the context of distributed power systems. A distributed power system comprises several sub‐networks, or zones with dedicated control and monitoring centers. The overall system is represented by linear time invariant state space models with coupled dynamical and algebraic equations. In the proposed strategy, the attacker has access to only one of the sub networks; therefore, the attacker only requires local information about one of the power system zones. Primarily, the proposed attack policy is defined based on zero‐dynamics of the sub network. The intruder injects predesigned signals to both the local generation unit controller as well as local unsecured and controllable loads in the attacked zone. Moreover, the local measurement system, or the sensors of the targeted zone are tampered. Furthermore, it will be proved that although the neighbor zones have physical connections with the attacked zone, the injected adversary signals are designed as they do not impact other zones directly in order to conceal the local attack from neighbor control centers as much as possible. We provide some advice to system administrators to make the intrusion unfeasible or to reveal the attack. The simulations on IEEE‐118 bus test system illustrate the validity of the assertions.

Discrete state space systems of fractional order

The present article devotes to the study of linear time invariant discrete fractional order state space systems using a novel approach. First, we replace the conventional Grünwald-Letnikov-type backward difference operator with the equivalent Riemann-Liouville-type nabla difference operator and obtain the system response using variation of constants and discrete Laplace transform methods. Next, we present three different algorithms to construct state transition matrix of the system. Finally, we provide an example to illustrate the applicability of established result.

Discrete state space systems of fractional order

The present article devotes to the study of linear time invariant discrete fractional order state space systems using a novel approach. First, we replace the conventional Grunwald-Letnikov-type backward difference operator with the equivalent Riemann-Liouville-type nabla difference operator and obtain the system response using variation of constants and discrete Laplace transform methods. Next, we present three different algorithms to construct state transition matrix of the system. Finally, we provide an example to illustrate the applicability of established result.

Model order reduction of commensurate linear discrete-time fractional-order systems

This paper presents an approach to model order reduction of linear time-invariant discrete-time commensurate fractional-order state space systems by means of the Balanced Truncation Approximation and Singular Perturbation Approximation methods. Mathematical formulas for computation of controllability and observability Gramians for the fractional-order systems are derived. This enables an extension of classical SVD-originated model order reduction algorithms to model reduction of discrete-time fractional-order state space systems. Simulation experiment illustrates the efficiency of the introduced methodology.

Iterative Residual Generator for Fault Detection With Linear Time-Invariant State–Space Models

In this paper, an iterative residual generator (IRG) is proposed for discrete time-invariant state-space model with the aim of detecting faulty signals. By minimizing the mean square errors subject to unbiasedness constraint, a new filter with finite impulse response structure is derived. The resulting IRG is then obtained by extracting residual signal from the batch filter through several predictor/corrector iterations. It shows that IRG can provide a zero-mean Gaussian process regardless of previous estimation errors. More importantly, it includes the residual generation process in the Kalman filter as its special case. With the chi-square test, a numerical example is simulated to demonstrate that IRG can reduce the false alarm significantly compared with the traditional recursive strategy in the presence of actuator or sensor faults, and the estimation horizon length in IRG serves as a tuning parameter providing a tradeoff between the missed alarm and false alarm.

Modeling of discrete-time fractional-order state space systems using the balanced truncation method

This paper presents a new approach to approximation of linear time-invariant (LTI) discrete-time fractional-order state space SISO systems by means of the SVD-originated balanced truncation (BT) method applied to an FIR-based representation of the fractional-order system. This specific representation of the system enables to introduce simple, analytical formulas for determination of the Cholesky factorizations of the controllability and observability Gramians, which contributes to significant improvement of the computational efficiency of the BT method. As a model reduction result for the fractional-order systems we obtain a low-order rational (integer-order) state space system. Simulation experiments show a high efficiency of the introduced methodology both in terms of the approximation accuracy of the model and low time complexity of the approximation algorithm.

Regional robust stabilisation and domain-of-attraction estimation for MIMO uncertain nonlinear systems with input saturation

ABSTRACTThe regional robust stabilisation by means of linear time-invariant state feedback control for a class of uncertain MIMO nonlinear systems with parametric uncertainties and control input saturation is investigated. The nonlinear systems are described in a differential algebraic representation and the regional stability is handled considering the largest ellipsoidal domain-of-attraction (DOA) inside a given polytopic region in the state space. A novel set of sufficient Linear Matrix Inequality (LMI) conditions with new auxiliary decision variables are developed aiming to design less conservative linear state feedback controllers with corresponding larger DOAs, by considering the polytopic description of the saturated inputs. A few examples are presented showing favourable comparisons with recently published similar control design methodologies.

Single Range Localization in 3-D: Observability and Robustness Issues

The problem of estimating the position of a 3-D vehicle subject to a constant unknown velocity disturbance is addressed: the only available model output is assumed to be the distance (range) to a reference point. The vehicle’s nominal velocity is also assumed to be known. An observability analysis is performed and an observer is designed. The proposed approach departs from alternative ones and leads to the definition of a linear time-invariant state equation with a linear time-varying output. The localization problem is solved using a novel outlier robust predictor–corrector state estimator. Numerical simulation examples are described to illustrate the performance of the method compared with a standard Kalman filter.

Combined state and multi‐innovation parameter estimation for an input non‐linear state‐space system using the key term separation

In this study, the authors study the state and parameter estimation problem for an input non-linear system consisting of a static non-linear block and a linear time-invariant state space subsystem. Using the filtering technique, a filtering based multi-innovation generalised stochastic gradient (SG) algorithm is proposed for avoiding estimating the redundant parameters based on the key term separation technique. Compared with the multi-innovation generalised SG algorithm, the proposed algorithm has higher parameter estimation accuracy. Two simulation examples are provided to show that the proposed algorithm works well.

An approach to design MIMO FO controllers for unstable nonlinear plants

This paper develops an approach to control unstable nonlinear multi-inputs multi-output (MIMO) square plants using MIMO fractional order (FO) controllers. The controller design uses the linear time invariant (LTI) state space representation of the nonlinear model of the plant and the diagonal closedloop transfer matrix (TM) function to ensure decoupling between inputs. Each element of the obtained MIMO controller could be either a transfer function (TF) or a gain. A TF is associated in turn with its corresponding FO TF. For example, a D (Derivative) TF is related to a FO TF of the form D±, ± = [0, 1]. Two applications were performed to validate the developed approach via experimentation: control of the angular positions of a manipulator, and control of the car and arm positions of a translational manipulator.

Gain Scheduling Dual Mode MPC for a Solar Thermal Power Plant

A nonlinear gain scheduling control strategy is proposed for a concentrated solar thermal power plant. The strategy involves the identification of local linear time-invariant state space models around a family of operating points, the design of corresponding local linear dual mode model-based predictive controllers and the selection of an appropriate scheduling variable to govern the switching. The local models are estimated directly from input-output data using a subspace identification method while taking into account the frequency response of the plant. Input-output data are obtained from a nonlinear simulation model of the plant rather than the plant itself. The effectiveness of the proposed control strategy in terms of tracking and disturbance rejection is evaluated through two different scenarios created in a nonlinear simulation environment.