Linear quadratic optimization for positive LTI system

This study investigates characteristic root equivalency relations between commensurate order and integer order Linear Time Invariant (LTI) systems. Author introduces some useful properties of a special class of commensurate order systems, which is called characteristic root equivalency class of LTI systems. These properties present potential to facilitate design and analysis efforts of this class of commensurate order systems. In this sense, straightforward stability checking procedures and design approaches for commensurate order root equivalent systems of the first and second order LTI systems are demonstrated. Findings of the study are validated by illustrative examples.

Frequency-domain stability conditions for asynchronously sampled decentralized LTI systems

In this paper, the cooperative stabilization problem of linear time-invariant (LTI) systems is studied based on a class of distributed Luenberger observers. A given LTI system is estimated by a network of observers. In case of observability configuration or external attack, each observer has to possess different measurement output of the LTI system. Classical distributed control strategies cannot be directly used because they are assumed to obtain identical information. In view of this problem, a class of distributed observers are designed under an undirected graph, which can recover the state of the LTI system by limited local information. By using any single channel of the estimated state, a control input is proposed to stabilize the LTI system. By analysing the stability of an augmented system which involves the LTI system and the estimation error system, some necessary and sufficient conditions are derived for the existence of control gains that guarantee the stabilization of the LTI system. In addition, some algorithms are proposed to specify the control gains. Lastly, some simulation examples are given to verify the effectiveness of the theoretical result.

The paper deals with the identification of linear time invariant (LTI) systems by a special observer. An observer emitting an frequency modulated continuous wave (FMCW) signal and having a stretch processor as receiver will be considered for system identification. A thorough derivation of the gathered baseband signal for arbitrary LTI systems will be given. It is shown, that the received signal is approximately given by the transfer function of the LTI system over the frequency sweep of the FMCW signal. The proof relies on an infinite large time-bandwidth product of the transmit signal, such that errors remain in practical applications with a finite time-bandwidth product. Monte–Carlo simulations are conducted to verify the approximation and to quantify its accuracy and remaining errors. The findings are important for e.g. calibration or derivation of a device model in FMCW radar applications.

Chapter 4 - Frequency Analysis: The Fourier Series

In this paper, we investigate two classes of consensus protocols for networked multi-agent systems: linear time-invariant (LTI) systems and linear time-delay systems. Based on the topology of multi-agent systems, the first-order integrator model is developed. The digraph (directed graph) is employed to show the topology of networked systems, and then a consensus convergence criterion is established. For LTI systems, we prove that their consensus value will converge globally asymptotically to the convex hull of initial states. By solving a set of linear equations, we get the convex combinations of equilibria, and we obtain the convex value of continuous system. If the topology is fixed and time-invariant, the consensus value of the linear time-delay system is also the convex hull of the initial states and is identical to the LTI system. Finally, a network of six agents is presented to show the effectiveness of the results of this paper.

Chapter 13 - LTI Systems: Convolution, Correlation, Coherence, and the Hilbert Transform

In incorporating an internal model (IM) to an linear time invariant (LTI) feedback system to achieve asymptotic regulation for exogenous disturbances, limitation in robust stability often hinders the transient response performance, such as the convergence rate and maximum output magnitude, of the LTI closed loop system. When the onset time of the disturbance introduced to the system is known, we formulate a linear time varying (LTV) controller, which exploits this information to improve the transient performance while maintaining the asymptotic regulation and desired robust stability. Simulation and experimental results for asymptotic regulation of a linear motor are presented to demonstrate the improvement of the proposed LTV over LTI controller.

Stepped chirp signal is widely employed in high resolution radar systems. Conventional spectrum reconstruction methods are sensitive to system errors. In this paper, a novel range profile synthesis algorithm is proposed based on multichannel blind deconvolution (MBD) theory. The received signals are modelled as the outputs of a series of linear time-invariant (LTI) systems with a common unknown input. In the presence of system errors, the impulse responses of the LTI systems are also unknown. Thus, range profile synthesis is formulated as a MBD problem. Regularizations are introduced to solve the ill-posedness. Alternating optimization method is used to solve the problem. Simulation results show that the proposed method is able to provide accurate range profile estimations. (4 pages)

In this paper, a MIMO PI design procedure is proposed for linear time invariant (LTI) systems with multiple time delays. The controller tuning is established in two stages and guarantees performances for set‐point changes, disturbance variations, and parametric uncertainties. In the first stage, an iterative linear matrix inequality (ILMI) approach is extended to design PI controllers for systems with multiple time delays without performance guarantee, a priori. The second stage is devoted to improve the closed‐loop performances by minimizing sensitivity functions. Simulations results carried out on the unstable distillation column, the stable industrial scale polymerization (ISP) reactor, and the non‐minimum phase 4‐tank benchmark prove the efficiency of the proposed approach. A comparative analysis with the conventional internal model control (IMC) approach, a multiloop IMC‐PI approach, and a previous ILMI PID approach proves the superiority of the proposed approach compared to the related ones.

In this paper, we present an output observer for single output linear time invariant (LTI) systems based on the temporal state solution of the system. The aim of an output estimator/observer is to reconstruct the output of the system for various purposes, namely fault detection, filtering and output stabilisation among others. A distinctive feature of this observer is that time variable is used as input in the observer equation. Proper transformations are employed to avoid the inversion of large matrices involved in the system's solution computation. Numerical simulations are presented with and without noise presence.

This paper reviews robust feedback control applications in power system. Robust control concept in power system control area evolves by the way of uncertainty modeling. Uncertainties can be caused by variation of loading, load parameters, network topology and machine parameters. One direct way to take into consideration of uncertainties is to enumerate various system operating conditions and use a set of linear time invariant systems to model the power systems. Enumeration method is used in robust control signal selection and robust controller design. Another way is to model uncertainties as structured or unstructured uncertainties. Controllers are designed based on nominal system condition to tolerate uncertainty as much as possible. The controller synthesis problem is solved by either Algebraic Ricatti Equation (ARE) or Linear Matrix Inequality (LMI). LMI is shown to be more flexible in controller design since it provides a way to specify system dynamic performance. Most recently, linear parameter varying (LPV) concept expands the use of enumeration method to express a set of linear time invariant (LTI) systems with state space dependent on linear parameters. LPV based control methods extend gain scheduling to design linear control for each operating conditions and overall a nonlinear controller for the varying parameter system.

In this paper, a closed loop subspace predictive control algorithm is developed for linear time invariant (LTI) systems with static nonlinearity at the plant input, i.e. for Hammerstein LTI systems (H-SPC). The elaborated method is a data driven approach combining subspace identification with optimal predictive control. Using the closed loop structure of the subspace identification technique an input-output predictor is derived with the estimated Markov parameters and with the identified static input nonlinearity. Identification makes use of the kernel of the data space by least square support vector machinery (LS-SVM). A nominal prediction is then created. The goal of the control is to optimize the associated quadratic performance criteria in a receding horizon fashion. A simulation example is provided that shows the effectiveness of the proposed methodology.

Two approaches to control system design for constrained systems are studied. The first involves theoretical investigations of constrained model predictive control algorithms. The second involves extensions of robust linear control theory to handle the nonlinear control schemes commonly used in practice for constrained systems. A novel model predictive control algorithm, with attractive functional and numerical characteristics is developed. This algorithm minimizes peak excursions in the controlled outputs and is particularly suited to regulatory control problems common in continuous process systems. Model predictive control concepts are extended to uncertain linear systems. An on-line optimizing control scheme (RMPC) is developed which has as its objective the minimization of worst-case tracking error for an entire family of linear plants. For model uncertainty descriptions which provide plant impulse response coefficients as affine functions of uncertain parameters, it is shown that the required minimax optimization problem can be recast as a single linear program. The discrete time optimal averaging level control problem is formulated and solved. A finite horizon approximation to the problem is introduced and analytical solutions are obtained in important special cases. A model predictive control formulation is introduced which provides optimal flow filtering and integral action. Analysis tools are provided to characterize the trade-off between flow filtering and rapid integral action. A complete theory is developed for the multivariable anti-windup, bumpless transfer (AWBT) problem. The theoretical framework allows the consideration of any linear time invariant (LTI) control system subject to plant input limitations and substitutions. A general AWBT compensation scheme, applicable to multivariable controllers of arbitrary structure and order, is developed. Conditions are derived under which this general AWBT method reduces to any one of several well-known heuristics for AWBT (e.g., PI anti-reset windup and IMC). The design issues which affect AWBT performance are identified and quantitative analysis methods are developed. Sufficient conditions for nonlinear stability of the AWBT compensated system are provided. These results are a generalization of, and are less conservative than, those available in the AWBT literature. The definition of AWBT performance objectives which are independent of controller structure allows the formulation of a general AWBT synthesis problem. This formal synthesiproblem addresses each of the identified performance objectives in a quantitative manner. The synthesis problem is shown to be a special case of a constrained structure controller synthesis (CSCS) problem. A solution method via reduction to static output feedback is presented and the engineering trade-offs available in the AWBT design are discussed.

Analytical Boundaries of Controller Gains for Delay-independent Stability of LTI Systems with Single Output Delay