NCS - 15b - Asymptotic stability, exponential stability, local and global stability

In the last lecture we talked about Lyapunov stability and in today's lecture we shall discuss a few more associated definitions that is we shall talk about asymptotic stability, exponential stability, local and global stability. So here is the definition of Lyapunov stability which we discussed in the last lecture that is an equilibrium point of this non-linear system is stable if this expression is satisfied. What is meaning of this expression? This expression means that corresponding to each value of epsilon there exists some non-zero value of delta such that trajectories starting within ball of radius delta these trajectories stay within ball of radius epsilon and this is satisfied for all initial conditions and for all values of epsilon. What is asymptotic stability? So an equilibrium point is asymptotically stable if it is stable that is it satisfies the definition of Lyapunov stability and in addition to that this expression is satisfied that is the trajectories finally converge to the equilibrium point at origin. This is graphically demonstrated over here if corresponding to each value of epsilon there exists delta such that trajectories starting from within ball of radius delta these trajectories stay within ball of radius epsilon then it will be stable and if in addition to that these trajectories finally converge to the equilibrium point at origin then this equilibrium point will be asymptotically stable. Let's demonstrate it with our favorite example of pendulum system if we drop pendulum from here it will oscillate about its equilibrium point at origin and if there is friction this pendulum will finally settle down at its equilibrium point which is at origin. Hence this system is asymptotically stable that is also apparent from the phase portrait of the pendulum system and we see that the trajectories finally converge to the equilibrium point at origin and a set of all initial conditions corresponding to which all trajectories converge to the equilibrium point that set is called region of attraction it is also called region of asymptotic stability or domain of attraction or basin. Here is the phase portrait of this pendulum system which we discussed earlier multiple equilibrium points are shown over here this is the equilibrium point at origin the equilibrium point in upright position and the equilibrium point at 2 pi 0 which means that this pendulum will settle down after one complete rotation to this point. So multiple equilibrium points are shown here in this graph if we talk about the equilibrium point at origin what we see is that for certain initial conditions trajectories converge to the this equilibrium point for others for example if we talk about this initial condition the trajectories do not converge to this equilibrium point. So set of all the initial conditions corresponding to which trajectories converge to the equilibrium point that set of initial condition is called region of attraction for this particular system here is the set and we see that all trajectories all starting from within this set those trajectories converge to the equilibrium point at origin. So this region indicated over here that is the region of attraction for this equilibrium point. What is exponential stability an equilibrium point is exponentially stable if the trajectories decay faster than an exponential term where lambda and k are some positive constants any positive constants. This is demonstrated over here in this graph here is the trajectory of a system and we see that this trajectory decays faster than this particular exponential term therefore the system corresponding to which this trajectory is plotted that system is exponentially stable. For example consider this non-linear system and we can verify that this is the solution to this non-linear differential equation and here what is this thing inside this brackets this sin is always positive. So 1 plus sin square x of tau that will always be greater than 1 and its integral its integral will always be greater than the integral of 1 that is this x of t is always less than this expression that is trajectories decay faster than an exponential term hence this equilibrium point of this non-linear system which is at origin is exponentially stable. What we additionally see from this graph is that this exponential term decaying exponential term will finally its final value will be 0 that is if an equilibrium point of a system is exponentially stable it will be asymptotically stable as well because the trajectories finally converge to the equilibrium point at origin. So exponential stability implies asymptotically stability. However the opposite is not true it is quite possible that a system is an equilibrium point of a system is asymptotically stable but it is not exponentially stable. Here is a counter example we have a non-linear system and the solution to this non-linear differential equation is given over here you can easily verify it that this expression satisfies this differential equation by taking the derivative of this expression and substituted substituting it over here. So what you can see from this expression is that as t tends to infinity x of t tends to 0 that is the equilibrium point of this system is asymptotically stable. However this expression decays slower than any exponential function that is we cannot find any positive value of lambda and this k over here such that this expression is satisfied for this particular system. Hence although this system is asymptotically stable however it is not exponentially stable. An additional observation is that for linear systems if a linear system is stable we exclude this marginal stability case. So if a linear system is stable then it is always exponentially stable it is asymptotically stable as well as exponentially stable that is apparent from this expression. If we consider a linear system its solution is given by this expression and we can see that we can find an exponential term such that the states converge faster than that exponential term. Hence linear systems if linear systems are stable then these are asymptotically stable as well as exponentially stable. What is a global stability and local stability? If asymptotic stability holds for any initial state then the equilibrium point is said to be globally asymptotically stable and if it holds only for certain set of initial conditions then that equilibrium point is called locally asymptotically stable or locally exponentially stable. Consider for example these two systems here we have a ball rolling on this surface and what we can see is that if there is friction then this ball will finally settle down at its equilibrium point over here. Hence this equilibrium point is asymptotically stable and furthermore wherever we place this ball on this surface it will finally settle down to this equilibrium point that is asymptotic stability holds for any initial condition. Hence the equilibrium point for this system is globally asymptotically stable. Compared to that here we have another system and it has multiple equilibrium points one equilibrium point is here, here is another equilibrium point and here is the third equilibrium point and we can see that this equilibrium point and this equilibrium point these are stable this is unstable equilibrium point. However this equilibrium point and this equilibrium point these are only locally stable equilibrium points. If we drop this ball from for example from this position it will finally settle down at this equilibrium point. However if we drop this ball from for example some position over here or position over here it will not settle down to this equilibrium point. So this equilibrium point is stable only for certain set of initial conditions hence this equilibrium point as well as this equilibrium point these are only locally asymptotically stable. An additional observation is that if a system has multiple equilibrium points then it cannot be globally asymptotically stable because there definitely exist certain initial conditions for which the trajectories will not converge to the equilibrium point under consideration. Hence global stability is only studied for systems with one equilibrium point. A further observation is that for linear systems if linear systems are stable then these are always globally asymptotically stable as well as globally exponentially stable. That is apparent from this expression here we have a linear system and solution to this linear system is given by this expression and we can see that whatever is the initial condition if this term is decaying the trajectories will finally converge to the equilibrium point at origin. So for all initial conditions trajectories will converge to the equilibrium point hence this linear system is globally asymptotically stable as well as globally exponentially stable. So these were a few additional definitions and in the next lecture we shall study the Lyapunov theorem that helps us to determine stability of equilibrium points for non-linear systems.