Low‐gain integral control for a class of discrete‐time Lur'e systems with applications to sampled‐data control

We study low-gain (P)roportional (I)ntegral control of multivariate discrete-time, forced Lur’e systems to solve the output-tracking problem for constant reference signals. We formulate an incremental sector condition which is sufficient for a usual linear low-gain PI controller to achieve exponential disturbance-to-state and disturbance-to-tracking-error stability in closed-loop, for all sufficiently small integrator gains. Output tracking is achieved in the absence of exogenous disturbance (noise) terms. Our line of argument invokes a recent circle criterion for exponential incremental input-to-state stability. The discrete-time theory facilitates a similar result for a continuous-time forced Lur’e system in feedback with sampled-data low-gain integral control. The theory is illustrated by two examples

Positive state controllability of positive linear systems

Controllability of positive systems by positive inputs arises naturally in applications where both external and internal variables must remain positive for all time. In many applications, particularly in population biology, the need for positive inputs is often overly restrictive. Relaxing this requirement, the notion of positive state controllability of positive systems is introduced. A connection between positive state controllability and positive input controllability of a related system is established and used to obtain Kalman-like controllability criteria. In doing so we aim to encourage further study in this underdeveloped area

A note on the eigenvectors of perturbed matrices with applications to linear positive systems

A result is presented describing the eigenvectors of a perturbed matrix, for a class of structured perturbations. One motivation for doing so is that positive eigenvectors of nonnegative, irreducible matrices are known to induce norms — acting much like Lyapunov functions — for linear positive systems, which mayhelp estimate or control transient dynamics. The results apply to both discrete- and continuous-time linear positive systems. The theory is illustrated with several examples

Make movement your mission: evaluation of an online digital health initiative to increase physical activity in older people during the COVID-19 pandemic

OBJECTIVE: To formatively evaluate the Make Movement Your Mission (MMYM) digital health initiative to promote physical activity (PA) levels and help avert the negative consequences of sedentary behaviours in older adults during the SARS-CoV2 pandemic. METHODS: Mixed-method study to explore activity levels, changes in physical function and Activities of Daily Living (ADLs), quality-of-life, social engagement, technology use, and accessibility. Survey data were analysed descriptively. Qualitative interviews were analysed using framework analysis. RESULTS: Forty-one respondents completed the survey (Mean age 68.4 (8.9) years; 34 Female), 68% aged ≥ 65 years. Average attendance was 14.3 sessions per week (3.5 h). 73% had been with MMYM for >1 year, 90% reported they were engaging in more movement on a typical day, and 75% reported improvement in ability to perform moderate PA. Since starting MMYM, participation in activities targeting strength, balance and flexibility increased (by 48%, 73% and 75%, respectively). 83% met strength and 90% balance PA guidelines for health (≥ 2x per week). Between 18% and 53% of respondents reported improvements in ADLs, 53% reported better quality-of-life, and 28% increased use of the internet. Eight participants were interviewed (Mean age 70.7 (6.7) years; 7 Female). Activity levels were promoted by having direct support from the instructor through Facebook messages pre and post live sessions, having group expectation about quality and level of engagement, having a sense of control and encouragement from others, MMYMs regularity, choice around level of engagement and accessibility. Noticing short-term outcomes in balance and posture helped boost confidence and continued participation. CONCLUSION: Clinical trials need to robustly assess its effectiveness and acceptability

A switching feedback control approach for persistence of managed resources

An adaptive switching feedback control scheme is proposed for classes of discrete-time, positive difference equations, or systems of equations. In overview, the objective is to choose a control strategy which ensures persistence of the state, consequently avoiding zero which corresponds to absence or extinction. A robust feedback control solution is proposed as the effects of different management actions are assumed to be uncertain. Our motivating application is to the conservation of dynamic resources, such as populations, which are naturally positive quantities and where discrete and distinct courses of management actions, or control strategies, are available. The theory is illustrated with examples from population ecology

The converging-input converging-state property for Lur’e systems

Using methods from classical absolute stability theory, combined with recent results on input-to-state stability (ISS) of Lur{\textquoteright}e systems, we derive necessary and sufficient conditions for a class of Lur{\textquoteright}e systems to have the converging-input converging-state (CICS) property. In particular, we provide sufficient conditions for CICS which are reminiscent of the complex Aizerman conjecture and the circle criterion and connections are also made with small gain ISS theorems. The penultimate section of the paper is devoted to non-negative Lur{\textquoteright}e systems which arise naturally in, for example, ecological and biochemical applications: the main result in this context is a sufficient criterion for a so-called “quasi CICS” property for Lur{\textquoteright}e systems which, when uncontrolled, admit two equilibria. The theory is illustrated with numerous examples

Robust set-point regulation for ecological models with multiple management goals

Population managers will often have to deal with problems of meeting multiple goals, for example, keeping at specific levels both the total population and population abundances in given stage-classes of a stratified population. In control engineering, such set-point regulation problems are commonly tackled using multi-input, multi-output proportional and integral (PI) feedback controllers. Building on our recent results for population management with single goals, we develop a PI control approach in a context of multi-objective population management. We show that robust set-point regulation is achieved by using a modified PI controller with saturation and anti-windup elements, both described in the paper, and illustrate the theory with examples. Our results apply more generally to linear control systems with positive state variables, including a class of infinite-dimensional systems, and thus have broader appeal

Stabilisation by adaptive feedback control for positive difference equations with applications in pest management

An adaptive feedback control scheme is proposed for stabilising a class of forced nonlinear positive difference equations. The adaptive scheme is based on so-called high-gain adaptive controllers, and contains substantial robustness with respect to model uncertainty as well as with respect to persistent forcing signals, including measurement errors. Our results take advantage of the underlying positive systems structure and ideas from input-to-state stability from nonlinear control theory. Our motivating application is to pest or weed control, and in this context the present work substantially strengthens previous work by the authors. The theory is illustrated with examples