Sample-time robustness of linear time-invariant multivariate hybrid systems

Frequency-domain stability conditions for asynchronously sampled decentralized LTI systems

In this paper, we analyse some fundamental structural properties of linear time-invariant multivariable systems in the controller canonical form and present a direct method for the computation of bases and associated friends for output-nulling, input-containing and reachability subspaces in terms of the parameters of the system and the invariant zero structure, both in the nondefective and in the defective case. Using this analysis, it is possible to express the solvability conditions of important control and estimation problems in terms of easily checkable conditions on the system matrices.

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

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.

This paper is concerned with the exact calculus of the switching times for linear time invariant hybrid systems. We study the system's behavior that incorporates dynamical switches from a current location to another and propose an explicit procedure for a constructive treatment of the switching times. An implementable computational method elaborated in this paper makes it possible to solve the multidimensional problem of switching times detections in an algorithmic form. In this sense, the above numerical procedure also determines the sequence of locations of the hybrid system. Finally, we apply the proposed constructive approach to two illustrative examples.

: This report considers a feedback control law for linear time-varying and time invariant discrete systems based on a receding horizon concept applied to a minimum energy problem with fixed terminal constraints. The control law is shown to be asymptotically stable and to result in a new method for stabilizing linear time-varying systems as well as extending some well known methods for stabilizing time invariant systems. In particular, the stabilizing gains of the feedback control law are obtained from the solution to a discrete Riccati equation over an arbitrary finite time interval, which is relatively easy to compute. The gain matrix reduces to a constant matrix for linear time invariant systems. Some stability results will turn out to be special cases of these results. The results parallel those for linear continuous time systems, although the technical details are tedious and more involved.

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

A matrix-operator representation of parameter sensitivities is used to provide an algebraic "structural" analysis of local parameter identifiability in linear time-invariant ordinary differential equation systems. Necessary conditions for identifiability depend only upon the system matrices, no integrals must be computed, and arbitrary parametrization may be used. Relations to insensitivity are discussed, and design techniques are suggested which use the nonidentifiable (insensitive) subspace to systematically reduce the number of exciting parameters in individually designed parameter identification experiments. Finally, sufficiency conditions for zero-state identifiability are examined in terms of control inputs which continually excite the natural modes of the parameter sensitivities.

Context: Exercise can cause fluctuations in blood glucose control in type 1 diabetics. For athletes with type 1 diabetes, maintenance of blood glucose within an ideal range may be difficult. Objective: To determine, in individuals with type 1 diabetes, the effectiveness of the closed loop control system versus the open loop control system in keeping blood glucose levels in the ideal range with exercise. Data Sources: A search of PubMed was conducted in June of 2020 using the Boolean phrases: (closed loop control system OR artificial pancreas) AND type 1 diabetes AND exercise AND ideal range AND adolescents, artificial pancreas AND glucose prediction AND exercise. Study Selection: Titles were reviewed for relevance, the abstract was then assessed for applicability, and finally the full text was examined. Articles were included that examined the percent of time in the ideal blood glucose range when exercise occurred during that day. Articles were excluded that didn’t compare the closed loop and open loop control systems and articles that did not involve exercise. Data Extraction: The PEDro scale was used to determine the methodological quality of the included studies. The measure addressed was the percent of time in the ideal blood glucose range of 70-180 mg/dL. 95% Confidence Intervals and Cohen’s D were calculated for each article. Data Synthesis: The search yielded 268 articles and 3 were selected for inclusion. The two randomized controlled trials scored 9/10 on the PEDro scale and the randomized two-arm crossover clinical trial scored 9/10 on the PEDro scale. Percent time spent in the ideal blood glucose range when exercise was performed was significantly higher in the closed loop group versus the open loop group in each of the three studies. In one randomized control trial, mean time in the ideal range was 71.3% (SD = 17.6, 95% CI = 62.5, 80.10) in the closed loop group versus 64.7% (SD = 13.3, 95% CI = 58.1–71.4) in the open loop group. Cohen’s D was 0.4. In the second randomized control trial, mean time in the ideal range was 73.5% (SD = 8.4, 95% CI = 70.1, 76.9) for the closed loop group versus 50% (SD = 26.8, 95% CI = 39.1, 60.9). Cohen’s D was 1.2. The two-arm crossover clinical trial resulted in a mean time in target range of 84.1% (SD = 11.5, 95% CI = 79.0, 89.2) in the closed loop group versus 68.7% (SD = 13.9, 95% CI = 62.5, 74.9) in the open loop group. Cohen’s D was 1.2. Conclusions: For adolescents with type 1 diabetes who exercise, the closed loop control system maintains blood glucose levels in the ideal range for a longer percent of time versus an open loop system. Each patient should be evaluated on a case-by-case basis with his/her healthcare team. Future research should examine the closed loop control system on specific energy systems.

S-54-2 - The integration of intrinsic properties and synaptic inputs in locus coeruleus

In this paper, Singular Perturbation Margin (SPM) and Generalized Gain Margin (GGM) are used as the classical Phase Margin (PM) and Gain Margin (GM) like stability metrics for multivariable Linear Time-Invariant (LTI) systems established from the view of the singular perturbation and the regular perturbation, respectively. The bijective relationship between the SPM and PM of a multivariable LTI system is established as corresponding lemmas and theorems, which is a generalisation of the single input single output cases. The analysis process for SPM and GGM are provided here. All the theoretical results are shown by a two-input-two-output LTI system (a simple model of a two-body spacecraft with non-collocated sensors and actuators using position feedback) as an application example. Such results are one essential step towards the big map of establishing a set of both theoretically rigorous and physically meaningful stability margin metrics for NLTV systems.

When high levels of cooled external exhaust gas recirculation (EGR) are used to increase engine efficiency, the laminar flame speed during the flame initiation period is reduced and the combustion duration is elongated. The reduction in flame propagation speed together with the increase in cycle-to-cycle variability makes the combustion process prone to failure of flame initiation (misfire) or slow burning rates. A highly visited operating speed/load point is used to investigate the feedback control of spark advance (SA) and EGR valve. A decentralized proportional–integral (PI) controller and a centralized linear quadratic Gaussian (LQG) controller are designed to maintain a desired combustion initiation and duration that indirectly ensures a proper flame kernel initiation and flame propagation. A simple control-oriented combustion model is derived from system identification to perform simulation and linear analysis of the closed-loop (CL) system. Experimental validation of the controllers shows that the LQG performs better in transients and produces the least amount of cycle-to-cycle variability in CL operation. Analysis of the linearized system in frequency domain shows that a multivariable architecture is required to handle the input–output coupling efficiently. Sensitivity analysis of PI and LQG controllers under gain variability is reported to guide the tuning process, which also influences the cycle-to-cycle variability. The two-input two-output controllers designed in this paper are compared with a single-input single-output (SISO) controller designed to only adjust SA for the maximum brake torque. When the EGR valve is adjusted based on a lookup table, uncertainty in the EGR-rate will directly affect the combustion duration for which the SISO controller alone cannot handle. In conclusion, a coordinated spark and EGR controller based on the measured combustion features is deemed to be required for combustion control under diluted conditions.

Nowaday the linear quadratic optimization subject to positive linear time invariant (LTI) system constitute an interesting study considering it can become a mathematical model of variety of real problem whose variables have to nonnegative and trajectories generated by these variables must be nonnegative. In this paper we propose a method to generate an optimal control of linear quadratic optimization subject to positive linear time invariant (LTI) system. A sufficient condition that guarantee the existence of such optimal control is discussed.

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)