The systems that can adapt in reaction to unanticipated external disturbances and events during their operation are called autonomous systems . These systems have capability of self-adaptation, self-configuration, self-monitoring, tuning, and other kinds of adjustment without requiring external control input . Autonomous systems are also referred to as self-managed systems due to their independence from external controls. These systems are built such that their components satisfy the design intent in the presence of small perturbations, external disturbances, noise, or other environmental changes.
In realm of control systems theory, autonomous systems carry the similar concept mentioned above. Talking in terms of state space model of dynamical control systems, the rate of change of states of the autonomous systems is independent of time. The dynamics of the autonomous system at any time instant will stay the same. The properties and behavior of the system is determined only from the individual states. The states represent trajectories when plotted as function of time. These trajectories at any instant should be the same as their past trend . It is possible that the autonomous functions depend on some kind of control input but it is necessary that the control input is itself function of states and overall closed loop system stays independent of time. If external control input is itself function of time in addition to state variables, then the closed loop system will be non-autonomous.
The notion of autonomous system will be clear with the help of real life examples. Consider the traditional mass spring system in which an object is attached to fixed supports with the help of spring and damper as shown below.
Figure 1: Mass Spring System
If the mass is at its equilibrium position, it will stay there forever until it is disturbed by an external force . Even when it is disturbed, it will return to its equilibrium position. It means that the system has the capability to adapt itself to external hazards and disturbances. Another popular example of autonomous systems is the swinging pendulum. The pendulum displaced to a certain position from its equilibrium point will continue to move to and fro until it reaches its motion is dampened and it returns to its equilibrium position.
The mathematical modeling of autonomous systems clarifies their characteristics. Using the state space modeling of dynamic systems, the mathematical definition of the autonomous system is :
A dynamic system with state space model of the form z't=fz,t is said to be autonomous if z't=fz. It means that the autonomous systems are time invariant and are function of states only. This implies that if z(t) represents the solution of the solution of the dynamic equation in some time interval (t1,t2), then x(t-α) will also be solution in interval (t1-α,t2-α) .
The transformation of non-autonomous systems to autonomous ones is quite simple and straightforward. The idea is to transform the time information into an additional state. For an n-dimensional non-autonomous system, the transformation towards autonomous system is achieved by taking xn+1'=1 as an additional dynamic equation. This results in (n+1)th state as xn+1=t-t0.This transformation clearly shows the strength of autonomous systems that their properties can be extended to time variant systems . However, the extension will result in loss of periodic solution .
The notion of state space is closely linked with trajectories in which the states the plotted with respect to each other. For autonomous systems, the trajectories don’t change being independent of time. If you can predict any state x1t from the information of a given state x0(t), then the prediction is also backwards applicable i.e. x0(t) can be predicted from x1(t) . Therefore, the biggest advantage autonomous systems possess over non-autonomous ones is their ease of analysis of phase plane portrait.
The dynamic equation can be split up into multiple states as follows:
z1't=f1(z1t, z2t,z3t,zn(t)
z2't=f2(z1t, z2t,z3t,zn(t)
The nth state equation will become:
zn't=fn(z1t, z2t,z3t,zn(t)
The dynamic equations of state clearly reflect the self-regulation nature of autonomous systems. The rate of change of states is the function of other states only and is independent of time. The future states can be predicted clearly from history provided we have the information of all states. The autonomous systems that are not controlled by any external input, they are same as time invariant systems. These systems may be linear or nonlinear, but don’t carry any dependency over time.
The stability analysis of autonomous systems can be conveniently carried out in terms of equilibrium points in phase plane portrait. Those points in state space at which the trajectory doesn’t evolve further are called equilibrium points. These points are stable if trajectories initiated nearby these points stay bounded. Asymptotic stability is guaranteed if the solutions starting near the equilibrium points move towards these equilibrium points. The concept of local stability near equilibrium points can also be extended to global stability. For rigorous stability analysis of autonomous systems, Lyapunov stability criteria are used. Lyapunov functions provided strong analysis tools for local and global stability of autonomous and non-autonomous systems.
In short, autonomous systems have time invariant characteristics and their dynamics are completely dependent on states only. The future states are predictable from present states and backward forecasting is also possible.
References
Khalil, H. K. (2002). Nonlinear systems, vol. 3. Prentice hall Upper Saddle River.
Kramer, J. a. (2008). Towards robust self-managed systems. Progress in Informatics, 1--4.
Polderman, J. W. (1998). Introduction to the mathematical theory of systems and contro. University of Twente, Department of Applied Mathematics.
Roxin. (1997). Control Theory and its Applications. Taylor & Francis.
Watson, D. P. (2005). Autonomous systems. Johns Hopkins APL technical digest, 368--376.
