# Resilience and Systems—A Review

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## Abstract

**:**

## 1. Introduction

## 2. Resilience—Roots and Evolution

## 3. Resilience Variants and Extensions

#### 3.1. Socio-Ecological and Engineering Resilience

… concentrates on stability near an equilibrium steady state, where resistance to disturbance and speed of return to the equilibrium are used to measure [resilience] …[71] (p. 33)

… emphasizes conditions far from any equilibrium steady state, where instabilities can flip a system into another regime of behavior—that is, to another stability domain. In this case, the measurement of resilience is the magnitude of disturbance that can be absorbed before the system changes its structure …[71] (p. 33)

… the capacity of a system to absorb disturbance and reorganize while undergoing change so as to still retain essentially the same function, structure, identity, and feedbacks …[42] (p. 1)

#### 3.2. Introduced Terminology

#### 3.3. Resilience-Inherent or Managed

#### 3.4. Resilience Engineering-Designing Resilience

… the intrinsic ability of a system to adjust its functioning prior to, during, or following changes and disturbances, so that it can sustain required operations under both expected and unexpected conditions…[111] (p. 36)

#### 3.5. Resilience and Sustainability-Related or Distinct Concepts

## 4. Systems Terminology

## 5. Resilience-Systems Adaptation

**Figure 5.**Changes to the system state and form: a graphical illustration of the resilience for a single dimensional system.

**Figure 6.**Block diagram illustration of a linear system state equations model with full-state feedback: an illustration for engineering resilience.

**Figure 7.**Block diagram illustration of a nonlinear system state equations model with full-state feedback: an illustration for socio-ecological resilience.

**Figure 8.**System active feedback along with its constituent elements: an illustration of resilience.

## 6. Conclusions and Future Research Directions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Number of publications with resilience in their titles—November 1973 to June 2022. Data source: Scopus [50].

**Figure 2.**Number of publications with resilience in their title, by discipline—November 1973 to June 2022. Data source: Scopus [50].

**Table 1.**Resilience journals/sources listing (highest to lowest) as per publications count—November 1973 to June 2022. Data source: Scopus [50].

Journal/Source | Publications Count |
---|---|

Sustainability Switzerland | 539 |

Ecology and Society | 297 |

PLoS ONE | 289 |

International Journal of Disaster Risk Reduction | 288 |

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | 278 |

International Journal of Environmental Research and Public Health | 277 |

Frontiers in Psychology | 208 |

IOP Conference Series: Earth and Environmental Science | 181 |

Reliability Engineering and System Safety | 147 |

Natural Hazards | 138 |

The rest of the 149 journals (including undefined journals) publishing fewer than a total of 135 publications per source | 6959 |

**Table 2.**Resilience authors listing (highest to lowest) as per publications count—November 1973 to June 2022. Data source: Scopus [50].

Author | Main Area(s) of Expertise | Publications Count |
---|---|---|

Ungar, M. | Social Works | 116 |

Pietrzak, R.H. | Clinical Psychology | 90 |

Linkov, I. | Risk and Decision Science | 85 |

Cimellaro, G.P. | Earthquake Engineering | 78 |

Southwick, S.M. | Psychiatry | 69 |

Masten, A.S. | Competence, Risk and Resilience | 59 |

Shaw, R. | Disaster risk and Climate Change | 58 |

Allen, C.R. | Ecological and Social-ecological Resilience | 53 |

Bonanno, G.A. | Psychology and Resilience | 51 |

Theron, L. | Educational Psychology | 50 |

Folke, C. | Social–ecological systems, Sustainability and Global Change | 46 |

Others | Various disciplines | <46 |

Author(s) | Publication Title | Citations Count |
---|---|---|

Holling [1] | Resilience and Stability of Ecological Systems | 19,670 |

Luthar et al. [52] | The Construct of Resilience: A Critical Evaluation and Guidelines for Future Work | 4284 |

Connor and Davidson [53] | Development of a New Resilience Scale: The Connor–Davidson Resilience Scale (CD-RISC) | 4043 |

Folke [54] | Resilience: The Emergence of a Perspective for Social–Ecological Systems Analyses | 3952 |

Masten [55] | Ordinary Magic: Resilience Processes in Development | 3817 |

Walker et al. [42] | Resilience, Adaptability and Transformability in Social–Ecological Systems | 3652 |

Bonanno [56] | Loss, Trauma, and Human Resilience: Have We Underestimated the Human Capacity to Thrive after Extremely Aversive Events? | 3483 |

Rutter [57] | Psychosocial Resilience and Protective Mechanisms | 2806 |

Lozupone et al. [58] | Diversity, Stability and Resilience of the Human Gut Microbiota | 2724 |

Hughes et al. [59] | Climate Change, Human Impacts, and the Resilience of Coral Reefs | 2648 |

Bruneau et al. [60] | A Framework to Quantitatively Assess and Enhance the Seismic Resilience of Communities | 2401 |

Norris et al. [61] | Community Resilience as a Metaphor, Theory, Set of Capacities, and Strategy for Disaster Readiness | 2378 |

Others | <2370 |

Terminology | Definitions | |
---|---|---|

System | An assemblage of functionally related components forming a unity whole to fulfill a certain purpose [121,122,123,124]. A system can be described by the fundamental variables of state, input, and output. | |

- Subsystem
| Components or layers of a system that collectively affect its behavior [121]. | |

- Environment
| That which is not part of a system is referred to as the environment and is separated from the system itself by its boundary, which is normally chosen as per the intent of the study [122]. | |

- $\mathrm{State}\left(\mathrm{x}\left(\mathrm{t}\right)\right)$
| $\mathrm{A}\mathrm{descriptor}\mathrm{of}\mathrm{the}\mathrm{system}\u2019\mathrm{s}\mathrm{internal}\mathrm{behavior}\mathrm{and}\mathrm{the}\mathrm{minimum}\mathrm{number}\mathrm{of}\mathrm{variables}(\mathrm{equal}\mathrm{to}\mathrm{the}\mathrm{order}\mathrm{of}\mathrm{the}\mathrm{system})\u2014\mathrm{known}\mathrm{as}\mathrm{the}\mathrm{state}\mathrm{variables}(\mathrm{also}\mathrm{referred}\mathrm{to}\mathrm{as}\mathrm{states})({\mathrm{x}}_{\mathrm{i}}\left(\mathrm{t}\right),\mathrm{i}=1,\dots ,\mathrm{n})$ that can completely represent the system and its behavior$\mathrm{to}\mathrm{a}\mathrm{certain}\mathrm{set}\mathrm{of}\mathrm{inputs}({\mathrm{u}}_{\mathrm{j}}\left(\mathrm{t}\right),\mathrm{i}=1,\dots ,\mathrm{m})$$.\mathrm{The}\mathrm{system}\mathrm{state}\mathrm{variables}\mathrm{are}\mathrm{not}\mathrm{always}\mathrm{directly}\mathrm{measurable}\mathrm{and}\mathrm{observable}.\mathrm{The}\mathrm{state}\mathrm{variables}\mathrm{collectively}\mathrm{constitute}\mathrm{the}\mathrm{state}\mathrm{vector}{\left[{\mathrm{x}}_{\mathrm{i}}\left(\mathrm{t}\right)\right]}^{\mathrm{T}}$ in the form of a unique point within the state space (also known as phase space) (Figure 5), where its evolution/trace over time is called the state trajectory and its graphical representation is labeled as the phase portrait [125,126]. | |

- Input
| The external forces acting upon the system are introduced in the form of inputs which are classified under two categories—those that are influenced by the engineer (referred to as controls) and those that cannot be influenced by the engineer [122]. | |

- Output
| Contrary to the system state, system output indicates the system’s external behavior, such as performance or response, and is normally a measure of direct interest to the engineer and is directly measurable and observable [122]. | |

- Perturbation
| A fundamental variable of the system which is related to the uncertainty in the system environment [90]. | |

State-space models | State-space models are formed by the interaction of the system’s fundamental variables (i.e., input, state, and output) and are generally given by the dual state and output equations. The coefficients used in the state equations to link the system fundamental variables are referred to as the system parameters [122]. State-space models are at the core of time-domain or state-space approaches—also known as the modern control theory and overcomes the apparent limitations of classical control theory (input-output transformations) by including the system state [125,127]. | |

- Linear models
| Linear models follow the supervision principle of linearity [128]. Equations (1) and (2) are continuous linear state-space equations for the system state and output, respectively. | |

$\dot{\mathrm{x}}\left(\mathrm{t}\right)=\mathrm{A}\mathrm{x}\left(\mathrm{t}\right)+\mathrm{B}\mathrm{u}\left(\mathrm{t}\right)$ | (1) | |

$\mathrm{y}\left(\mathrm{t}\right)=\mathrm{C}\mathrm{x}\left(\mathrm{t}\right)+\mathrm{D}\mathrm{u}\left(\mathrm{t}\right)$ | (2) | |

A—is the system, state, or dynamic matrix of (n × n) dimension, B—is the input or control matrix of (n × m) dimension, C—is the output matrix of (p × n) dimension, and D—is the direct transfer or feedforward matrix of (p × m) dimension, which is normally the zero or null matrix (Figure 6). | ||

- Nonlinear models
| Nonlinear models lack superposition properties but are a more accurate representation of most real-world problems despite their complexity.$\mathrm{A}\mathrm{general}\mathrm{state}\mathrm{equation}\mathrm{model}\mathrm{for}\mathrm{a}\mathrm{nonlinear}\mathrm{time}-\mathrm{variant}\mathrm{system}\mathrm{can}\mathrm{be}\mathrm{expressed}\mathrm{as}\mathrm{the}\mathrm{dual}\mathrm{of}\mathrm{state}(\mathrm{Equation}(3\left)\right)\mathrm{and}\mathrm{output}(\mathrm{Equation}(4\left)\right)\mathrm{equations}\mathrm{where}\mathrm{fi}\mathrm{and}\mathrm{gj}\mathrm{are}\mathrm{scalar}\mathrm{arguments}\mathrm{of}\mathrm{the}\mathrm{state}({\left[{\mathrm{x}}_{1},{\mathrm{x}}_{2},\dots ,{\mathrm{x}}_{\mathrm{n}}\right]}^{\mathrm{T}})$$,\mathrm{input}({\left[{\mathrm{u}}_{1},{\mathrm{u}}_{2},\dots ,{\mathrm{u}}_{\mathrm{m}}\right]}^{\mathrm{T}}),$and time (t) vectors (for the time-invariant case, the term t is absent in the model) [129,130] (Figure 7). | |

$\dot{\mathrm{x}}\left(\mathrm{t}\right)=\mathrm{f}\left[\mathrm{x}\left(\mathrm{t}\right),\mathrm{u}\left(\mathrm{t}\right),\mathrm{t}\right)]=\left[\begin{array}{c}\begin{array}{c}{\mathrm{f}}_{1}\left(\mathrm{x}\left(\mathrm{t}\right),\mathrm{u}\left(\mathrm{t}\right),\mathrm{t}\right))\\ {\mathrm{f}}_{2}\left(\mathrm{x}\left(\mathrm{t}\right),\mathrm{u}\left(\mathrm{t}\right),\mathrm{t}\right))\\ \vdots \end{array}\\ {\mathrm{f}}_{\mathrm{n}}\left(\mathrm{x}\left(\mathrm{t}\right),\mathrm{u}\left(\mathrm{t}\right),\mathrm{t}\right))\end{array}\right]$ | (3) | |

$\dot{\mathrm{y}}\left(\mathrm{t}\right)=\mathrm{g}\left[\mathrm{x}\left(\mathrm{t}\right),\mathrm{u}\left(\mathrm{t}\right),\mathrm{t}\right)]=\left[\begin{array}{c}\begin{array}{c}{\mathrm{g}}_{1}\left(\mathrm{x}\left(\mathrm{t}\right),\mathrm{u}\left(\mathrm{t}\right),\mathrm{t}\right))\\ {\mathrm{g}}_{2}\left(\mathrm{x}\left(\mathrm{t}\right),\mathrm{u}\left(\mathrm{t}\right),\mathrm{t}\right))\\ \vdots \end{array}\\ {\mathrm{g}}_{\mathrm{p}}\left(\mathrm{x}\left(\mathrm{t}\right),\mathrm{u}\left(\mathrm{t}\right),\mathrm{t}\right))\end{array}\right]$ | (4) | |

- Time-variant and time-invariant models
| If dynamic system parameters remain static and are not changed by the passage of time (only fundamental variables change), the system is called time-invariant; otherwise, the system is referred to as a time-variant system (e.g., adaptive systems) [122,125]. | |

- Continuous and discrete models
| For continuous in time (or space) models, the system fundamental variables are defined for all points of the independent variable(s) and described by differential equations, while for discrete in time (or space), the system fundamental variables are only defined at fixed points of the independent variable(s), and the models are described by difference equations. Continuous systems can be readily converted to discrete systems through the use of a proper discretization process, but the reverse is not feasible [122,125]. | |

- Deterministic and probabilistic models
| State-space approaches apply both to deterministic and probabilistic (stochastic) dynamical systems where the latter includes element(s) of randomness while the former does not. Whenever possible, deterministic state equations models are preferred over probabilistic ones for the sake of their simplicity, as well being less rigorous [122,125]. | |

Stability | The stability of dynamic systems may fall under the two broad categories of dynamic stability and structural stability. | |

- Dynamic stability
| Dynamic stability of a system is determined by matrix A (including the Jacobian matrix for nonlinear systems) in Equation (1) and is a measure of the tendency of a system’s state to return to its equilibrium (original state) or another suitable system state after being perturbed (in the absence of active feedback). Dynamic stability is equivalent to the system return rate (and/or settling time for a time-varying dominant eigenvalue) to the equilibrium which is measured by the dominant eigenvalue horizontal distance (real part) from the imaginary axis in the complex plane. A dominant eigenvalue/pole is related to the slow-moving state of the system and is closest located to the imaginary axis, which corresponds to the slowest and dominant decay/return rate to the equilibrium [131]. | |

- Structural stability
| Structural stability is indicative of keeping the system’s original form or another suitable system form (e.g., preventing bifurcations) after being perturbed by the changes within the system structure [132,133]. | |

Control action or system feedback | The control action is used for the controllability of the system state vector and is broadly categorized under the open-loop and closed-loop control actions. For linear systems, the control action (Equation (6)) entails both the control law- using matrix H (Equation (5)) and control matrix B (Equation (6)). | |

$u\left(t\right)=-\mathrm{H}\mathrm{x}\left(\mathrm{t}\right)$ | (5) | |

$\dot{\mathrm{x}}\left(\mathrm{t}\right)=\mathrm{A}\mathrm{x}\left(\mathrm{t}\right)+\mathrm{B}\left(\mathrm{H}\mathrm{x}\left(\mathrm{t}\right)\right)=\left(\mathrm{A}-\mathrm{BH}\right)\mathrm{x}\left(\mathrm{t}\right)$ | (6) | |

For nonlinear systems (Equation (3)), commonly, a local control action is introduced through the system linearization process around an operating state (original state) and, subsequently, a linear control action is used (Equation (6)). For global behavior of the nonlinear systems, control actions such as Control Lyapunov Functions (CLF) (full state-based control) (Figure 7) and Model Predictive Control (MPC) (output-based control) are used [134,135]. | ||

- Open-loop control
| Also known as feedforward control or passive control, where the control action is independent of the system state/output and is selected upfront [10]. | |

- Closed-loop control
| Also known as active feedback and is selected based on the monitoring of the system state/output and its subsequent comparison with a target (reference/equilibrium/steady-state) with the help of a control law or objective function [10] (Figure 8). Closed-loop control falls under three broad categories of optimal, robust, and adaptive controls. | |

- -
- Optimal control
| A control method to ensure state/output/system optimization around a reference point/path [136]. | |

- -
- Robust control
| The control law does not change over time for a certain range of the system form’s changes (a certain range of parameter uncertainties of the model) and is designed to optimize stability within a particular domain [136,137]. | |

- -
- Adaptive control
| The control law does change over time for the system form’s changes (parameter uncertainties of the model) and is designed to optimize stability for a certain criterion [138]. | |

Three main engineering problem-solving categories | The majority of real-world engineering problems fall under three fundamental system configurations: analysis, synthesis, and investigation [122]. | |

- Analysis
| Finding outputs from input and system model [122]. | |

- Synthesis
| Finding inputs from output and system model [122]. | |

- Investigation
| Obtaining a system model by using inputs and outputs [122]. |

Resilience Terminology | Equivalent Modern Control Systems (State-Space) Terminology | |
---|---|---|

$\mathrm{System}\mathrm{state}\mathrm{vector}{\left[{\mathrm{x}}_{\mathrm{i}}\left(\mathrm{t}\right)\right]}^{\mathrm{T}}$ | General system constituent components Terminology | |

$\mathrm{System}\mathrm{output}[\mathrm{y}\left(\mathrm{t}\right)]$ | ||

$\mathrm{System}\mathrm{original}\mathrm{state}\left[{\mathrm{x}}_{\mathrm{e}}\left(\mathrm{t}\right)\right]$$\mathrm{or}\mathrm{another}\mathrm{suitable}\mathrm{state}\left[{\mathrm{x}}_{\mathrm{s}}\left(\mathrm{t}\right)\right]$ | ||

- System identity or structure [145]
| Structural stability | |

- System behavior [31]
| System state trajectory | |

System state deviation around an objective function (synthesis treatment) | ||

System state vector dimensions/ranges on the phase-space/state-space | Passive feedback terminology | |

Dynamic stability—determined by matrix A (Equation (1)) or Jacobian matrix J, e.g., eigenvalues | ||

Subsystem’s interaction | ||

Nonlinear state-space models Complexity Bifurcations Stability radius | ||

Preset control | ||

System state—return to its original or another suitable state | ||

Robust closed-loop control | Active feedback terminology | |

Adaptive closed-loop control Time-variant state-space models | ||

Controllability |

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Mayar, K.; Carmichael, D.G.; Shen, X.
Resilience and Systems—A Review. *Sustainability* **2022**, *14*, 8327.
https://doi.org/10.3390/su14148327

**AMA Style**

Mayar K, Carmichael DG, Shen X.
Resilience and Systems—A Review. *Sustainability*. 2022; 14(14):8327.
https://doi.org/10.3390/su14148327

**Chicago/Turabian Style**

Mayar, Khalilullah, David G. Carmichael, and Xuesong Shen.
2022. "Resilience and Systems—A Review" *Sustainability* 14, no. 14: 8327.
https://doi.org/10.3390/su14148327