What do you understand by stability analysis?

Stability analysis is a part of the system and control theory, and it is used to understand or expect the stability of a system with the help of a mathematical model. It indicates how a model reacts to changes and perturbations.

The stability theory gives information about the stability of trajectories of dynamic systems and solutions of differential equations along with changes in initial conditions.
The heat equation is an equation that gives an idea about the variation in temperature of the system concerning time with changes in its initial condition as a result of the maximum principle.

Why do we need stability analysis?

The stability analysis is used to highlight the risks associated with system instability. It also helps in effective understanding and eliminating different systems' instabilities.

Overview in dynamical systems

Different elements of the qualitative theory of differential equations and dynamical systems deal with the limiting behavior of properties of solutions and trajectories with time.
The easiest way to represent the behavior of a system is displayed by equilibrium points, fixed points, and by periodic orbits. If a certain orbit of the system is known, it can be estimated easily whether a small change in the initial condition will lead to similar behavior or not. The stability theory addresses the following problems:

• Will a nearby behavior orbit indefinitely stay close to an equilibrium orbit?
• Will a nearby behavior orbit converge to or depart from an equilibrium orbit?

If a nearby orbit indefinitely stays close to a given orbit, the orbit is called stable. In contrast, if the orbit converges to the given orbit, it is called asymptotically stable, while the given orbit is said to be attractive in nature.

The stability of a system means that the trajectories do not change too much under small changes in the system's initial condition.
In contrast, it is also important where a nearby orbit is getting repelled from the given orbit. Generally, perturbing the initial state in some directions result in the trajectory asymptotically approaching the given direction.
While perturbing the initial state in other directions, the trajectory gets away from it. In some cases, the directions for which the behavior of the perturbed orbit is more complicated, neither a converging nor a escaping. The stability theory does not give sufficient information about the dynamics for those directions.

The stability theory also plays an important role in analyzing the qualitative behavior of an orbit under perturbations by using the linearization of the system near the orbit.
At each equilibrium of a smooth dynamical system with an n-dimensional phase, there is a specific n x n matrix A, whose eigenvalues represent the behavior of the nearby points (Hartman – Grobman theorem).
Additionally, if all eigenvalues are negative real numbers or complex numbers with negative real parts, the point is a stable fixed point, and the nearby points converge at an exponential rate. And suppose none of the eigenvalues are purely imaginary (or zero). In that case, the attracting and repelling directions are related to the eigenspaces of the matrix A with eigenvalues whose real part is negative and positive, respectively.

Stability analysis of a linear system

Consider a dynamical system of two first-order ordinary differential equations;

$$\begin{gathered} \stackrel{·}{x}=f\left(x, y\right) \\ \stackrel{·}{y}=g\left(x, y\right) \end{gathered}$$

Let $x_0$ and $y_0$ denote fixed points;

Let $x_0=y_0=0$, then;

$$\begin{gathered} f\left(x_0, y_0\right)=0 \\ g\left(x_0, y_0\right)=0 \end{gathered}$$

Then expand about $\left(x_0, y_0\right)$we will get,

$$\begin{gathered} \delta \stackrel{·}{x}=f_x\left(x_0, y_0\right)\delta x+f_y\left(x_0, y_0\right)\delta y+f_{xy}\left(x_0, y_0\right)\delta x\delta y+\dots \\ \delta \stackrel{·}{x}=g_x\left(x_0, y_0\right)\delta x+g_y\left(x_0, y_0\right)\delta y+g_{xy}\left(x_0, y_0\right)\delta x\delta y+\dots \end{gathered}$$

We can rewrite it as,

$\frac{d}{dt}\left[\frac{\delta x}{\delta y}\right]=\left[\begin{array}{cc}{f}_x\left(x_0, y_0\right)& f_y\left(x_0, y_0\right)\\ g_x\left(x_0, y_0\right)& g_y\left(x_0, y_0\right)\end{array}\right]\left[\begin{array}{c}\delta x\\ \delta y\end{array}\right];$

where the 2x2 matrix is called the stability matrix.

Now consider an n-dimensional map $x^{\text{'}}=T\left(x\right)$,

Let $x_0$ be a fixed point, so that

$T\left(x_0\right)=x_0$

Thus,

$$\begin{gathered} T\left(x_0+\delta x\right)=T\left(x_0\right)+\frac{\partial T}{\partial x}\delta x+O{\left(\delta x\right)}^2 \\ =T\left(x_0\right)+\delta T \\ or, \delta T=\frac{\partial T}{\partial x}\delta x\equiv A\delta x \end{gathered}$$

Now, this map can be transformed into the principal axis frame by finding the eigenvectors and eigenvalues of the matrix A,

$\left(A-\lambda I\right)\delta x=0$

So that the determinant,

$\left|A-\lambda I\right|=0$

Now, the mapping is

$\delta x_{prin{c}^{\text{'}}}=\left[\begin{array}{ccc}{\lambda }_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & {\lambda }_n\end{array}\right]$

When iterated a large number of times, $\delta T_{princ}^{\text{'}}\to 0$ only if $R\left[{\lambda }_i\right]<0$ for all $i$, but $\delta T_{princ}^{\text{'}}\to \infty$if any $R\left[{\lambda }_i\right]>0$. Analysis of the eigenvalues (and eigenvectors) of $A$, therefore, characterizes the type of fixed point.

Stability analysis of non-linear systems using Lyapunov Theory – I

Motivation

• The eigenvalue analysis concept cannot be useful for nonlinear systems because nonlinear systems have multiple limit cycles and stability points.
• The stability behavior of nonlinear systems is not same as linear systems.
• There is a specific condition for a systematic approach exploited for control design.

Linear stability analysis of continuous time nonlinear systems

Step-1: Find an equilibrium point of the system you are interested in.

Step-2: Calculate the Jacobian matrix of the system at the equilibrium point.

Step-3: Calculate the eigenvalues of the Jacobian matrix.

Now, if the real part of the dominant eigenvalue is:

Equal to zero: The equilibrium point may be neutral or we can say it as Lyapunov Stable.

Lesser than zero: The equilibrium point is stable.

Greater than zero: The equilibrium point is unstable.

Note that if other eigenvalues have real parts less than zero, the equilibrium point is a saddle point.

In addition, if complex conjugate eigenvalues are involved, oscillatory dynamics are going on around the equilibrium point. The equilibrium point is called a stable or unstable spiral focus if complex conjugate eigenvalues are dominant.

Common Mistakes

The most common mistake regarding stability analysis is the wrong input or wrong observation. Stability analysis is completely dependent upon the mathematical model. Now if you don't make the mathematical model correctly, the stability analysis will go wrong. 

Context and Applications

This topic is significant in the professional exams for both graduate and postgraduate courses, especially for

• Bachelor of Technology in Mechanical Engineering
• Bachelor of Science in Mathematics
• Master of Technology in Mechanical Engineering

Related Concepts

• Control systems
• Non-linear equations
• Analysis and model assessment
• Types of mathematical model

Practice Problems

Q1. Which of the following statement/statements is/are correct?

1. The effect of feedback is to reduce the system error.
2. Feedback increases the gain of the system in one frequency range but decreases in the other.
3. Feedback can cause an originally unstable system to become stable.
4. Both a and c.

Correct Option: (d)

Explanation: Feedback reduces error of the system by analyzing system's output, it can cause a stable system to become unstable or an unstable system to stable.

Q2. In which year the Routh-Hurwitz theorem was proved:

1. 1985
2. 1895
3. 1890
4. None of the Above

Correct Option: (b)

Explanation: The Routh-Hurwitz theorem is helpful in explaining the stability of a system without completely solving it. In 1895, John Routh and Adolf Hurwitz presented this theorem.

Q3. Which of the following option shows the effect of negative feedback in a system.

1. Increase in distortion noise
2. Increases the overall gain
3. Both A and B
4. None of these

Correct Option: (d)

Explanation: The negative feedback in a particular system decreases the noise of distortion as well as overall gain. So, option (d) is the correct answer.

Q4. Which of the following statement is correct for stability of system?

1. Small changes in the system input do not result in a large change in system output.
2. Small changes in the system parameters do not result in a large change in system output.
3. Small changes in the initial conditions do not result in a large change in system output.
4. All of the above mentioned.

Correct Option: (d)

Explanation: A system is said to be stable if small changes in its parameters or initial conditions does not result in a large change in system output.

Q5. In linear stability analysis of continuous-time nonlinear systems, if the real part of the dominant eigenvalue of the Jacobian matrix is greater than zero, then the equilibrium point is?

1. Stable
2. Unstable
3. Neutral
4. Cannot be determined

Correct Option: (b)

Explanation: In linear stability analysis of continuous-time nonlinear systems, if the real part of the dominant eigenvalue of the Jacobian matrix is,

Equal to zero: The equilibrium point may be neutral (Lyapunov Stable);

Lesser than zero: The equilibrium point is stable.

Greater than zero: The equilibrium point is unstable.

Note: If other eigenvalues have real parts less than zero, the equilibrium point is termed as a saddle point.