4. Definition of stability and algebraic stability criterion

4.1 The basic concepts of stability and the necessary and sufficient conditions for the stability of linear systems

Stability is an important performance of the control system, and it is also the primary condition for the normal operation and work of the system. The first prerequisite for the application of the control system in practical applications is that the system must be stable. The analysis of various quality indicators of the system must also be carried out under the premise of system stability.
In the actual operation of the control system, it is always disturbed by some external and internal factors, such as fluctuations in load and energy, changes in system parameters, and changes in environmental conditions. These factors always exist. If these factors are not considered in the system design and the designed system is unstable, then such a system is unsuccessful and needs to be redesigned or adjusted certain parameters or structures.
If the system is unstable, it will deviate from the original equilibrium state due to any slight disturbance, and diverge over time. Therefore, how to analyze the stability of the system and propose measures to ensure the stability of the system is one of the basic tasks of automatic control theory.
Other: A feedback condition is either stable or unstable-absolute stability. A system with absolute stability is called a stable system; if a closed-loop system is stable, the relative stability can be used to further measure its stability. For example: the more stable the aircraft, the more difficult it is to operate. However, the relative instability of modern fighter jets results in good maneuverability. Therefore, fighter jets are not as stable as commercial transport aircraft, but they can achieve rapid maneuverability.

4.2 Basic concepts of stability

Definition 1 : The system is in a certain initial equilibrium state. Under the influence of external disturbance, it deviates from this equilibrium state. When the external effect disappears, if enough time passes, the system can be restored to its original initial equilibrium state, then such a system is called a stable system. Otherwise, it is an unstable system.

Unstable example:

1. The closer the distance between the speaker and microphone of the studio sound system, the greater the echo, to a certain extent, the howling overwhelms the sound; (similar phenomena include telephones, computer microphones, etc.);

The stability is reflected in the nature of the transition process after the interference is eliminated. The deviation of the system from the equilibrium state can be considered as the initial deviation of the system.

Definition 2 : If the control system is under the effect of a sufficiently small initial deviation, the deviation of its transition process gradually tends to zero over time, that is, the system has the ability to restore the original equilibrium state, then the system is said to be stable; otherwise it is unstable .

Judgment method : judge whether all the poles of the transfer function are on the $s$ left half plane, or equivalently, judge whether the eigenvalues $A$ of the system matrix are all on the $s$ left half plane. If all the poles (or eigenvalues) are located in the $s$ left half plane, the relative stability of the poles (or eigenvalues) can be further judged .

Precautions:

1. Stability is the inherent property of the control system itself. This stability depends on the inherent characteristics (structure, parameters) of the system and has nothing to do with the input signal of the system;

A: For linear systems, the system is stable in a wide range (independent of input deviation);

B: For the actual "small deviation linearization" approximate linear system, the system is no longer stable after the deviation reaches a certain range.

2. Stability refers to the stability under free oscillation, that is, the stability of the system when the input is zero and the initial deviation is not zero; that is, it is the discussion of whether the free oscillation converges or diverges.

4.3 The necessary and sufficient conditions for system stability

Suppose the differential equation of the system or link is:
$\tiny y^{\left ( n \right )}\left ( t \right )+a_{n-1}y^{\left ( n-1 \right )}\left ( t \right )+\cdots +a_{1}\dot{y}\left ( t \right )+a_{0}y\left ( t \right )=b_{m}x^{\left ( m \right )}\left ( t \right )+b_{m-1}x^{\left ( m-1 \right )}\left ( t \right )+\cdots +b_{1}\dot{x}\left ( t \right )+b_{0}x\left ( t \right )$

In the formula: $\tiny x\left ( t \right )$ ——input, $\tiny y\left ( t \right )$ ——output $\tiny a_{i},\left ( i=0,\cdots,n-1 \right )$ ,; $\tiny b_{j},\left ( j=0,\cdots,m \right )$ is a constant coefficient.

Taking the above formula for a pull transformation, we get:

$\tiny \left ( s^{n}+a_{n-1}s^{n-1}+\cdots +a_{1}s+a_{_{0}} \right )Y\left ( s \right )=\left ( b_{m}s^{m}+b_{m-1}s^{m-1}+\cdots +b_{1}s+b_{_{0}} \right )X\left ( s \right )$

That is: $\tiny Y\left ( s \right )=\frac{\left ( b_{m}s^{m}+b_{m-1}s^{m-1}+\cdots +b_{1}s+b_{_{0}} \right )}{\left ( s^{n}+a_{n-1}s^{n-1}+\cdots +a_{1}s+a_{_{0}} \right )}\times X\left ( s \right )$

The function of the disturbance signal is equivalent to an impulse response signal $\tiny \delta \left ( t \right )$ to the system, and the output increment (deviation value) of the system is the response of the impulse input $\tiny C\left ( t \right )$ .

For the control system with the above transfer function, the pull transformation of pulse input is 1, that is $\tiny x_{i}\left ( s \right )=1$ , the pull transformation of the system output increment is:
$\tiny C\left ( s \right )=\frac{\left ( b_{m}s^{m}+b_{m-1}s^{m-1}+\cdots +b_{1}s+b_{_{0}} \right )}{\left ( s^{n}+a_{n-1}s^{n-1}+\cdots +a_{1}s+a_{_{0}} \right )}\times x_{i}\left ( s \right )=\sum_{j=1}^{n_{1}}\frac{a_{j}}{s+p_{j}}+\sum_{i=1}^{n_{2}}\frac{\beta _{i}\left ( s+\xi _{i}\omega _{i} \right )+\gamma _{i}\omega_{i}\sqrt{1-\xi _{i}^{2}}}{s^{2}+2\xi _{i}\omega _{i}s+\omega_{i}^{2}}$

$\tiny c\left ( t \right )=\sum_{j=1}^{n_{1}}e^{-p_{j}t}+\sum_{i=1}^{n_{2}}\beta _{i}e^{-\xi _{i}\omega _{i}t}cos\omega _{i}\sqrt{1-\xi ^{2}}t+\sum_{i=1}^{n_{2}}\gamma _{i}e^{-\xi _{i}\omega _{i}t}sin\omega _{i}\sqrt{1-\xi ^{2}}t$

The necessary and sufficient conditions for linear system stability:

The roots of the characteristic equation of the system (that is, the poles of the transfer function) are all negative real numbers or conjugate complex roots with negative real parts. In other words, the roots of the characteristic equation should all lie in the $s$ left half plane.

If there is a pair of conjugate complex roots whose real part is positive in the characteristic equation, its corresponding term is a divergent periodic oscillation;
If there is a positive real root in the characteristic equation, its corresponding exponential term will increase monotonically with time.

The system is unstable in the above two cases.

If there is a real root in the characteristic equation, which corresponds to a constant term, the system can be balanced in any state, which is called the random equilibrium state;
If there is a pair of conjugate imaginary roots in the characteristic equation, corresponding to periodic oscillations of equal amplitude, it is called the critical equilibrium state (or critical stable state).

From the perspective of control engineering, the critical stable state and the random equilibrium state are considered unstable.

note:

Stability is an attribute of the linear time-invariant system, which is only related to the structural parameters of the system itself, has nothing to do with the input and output signals, and has nothing to do with the initial conditions; it is only related to the poles, and has nothing to do with the zeros.

1. For the first-order system:

$\tiny a_{1}s+a_{0}=0$ ，

$\tiny s_{1}=-\frac{a_{0}}{a_{1}}$

As long as $\tiny a_{1},a_{0}$ both are greater than zero, the system is stable.

2. For the second-order system:

$\tiny a_{2}s^{2}+a_{1}s+a_{0}=0$ ，

$\tiny s_{1,2}=\frac{-a_{1}\pm \sqrt{a_{1}^{2}-4a_{2}a_{0}}}{2a_{2}}$

Only when $\tiny a_{2},a_{1},a_{0}$ both are greater than zero, the system is stable (negative real root or negative real part).

For third-order or above systems, finding roots is very tedious. So there is the following algebraic stability criterion.

4.4 Criteria for Algebraic Stability

4.4.1 Routh's stability criterion

Suppose the closed-loop characteristic equation of the linear system is:

$\tiny a_{0}s^{n}+a_{1}s^{n-1}+\cdots +a_{n-1}s+a_{n}=0$
The conditions for the stability of the system are:
a. The coefficients of the characteristic equation are $\tiny a_{i}\left ( i=0,\cdots ,n-1 \right )$ not equal to zero;

b. The signs of the coefficients $\tiny a_{i}$ of the characteristic equation are the same.

These two are necessary conditions.

E.g:

$\tiny q\left ( s \right )=\left ( s+2 \right )\left ( s^{2}-s+4 \right )=s^{3}+s^{2}+2s+8$

Sufficient condition: All items in the first column of the Routh permutation matrix composed of the coefficients of the characteristic equation are positive.
Necessary and sufficient conditions: For a stable system, there should be no sign changes in the first column of the Routh array.

This process needs to continue until the nth row is calculated, and the complete array of coefficients presents an inverted triangle.

note:

To simplify the calculation, a positive integer can be used to divide or multiply an entire row without changing the stability conclusion.

Routh's stability criterion:

The Routh stability criterion is based on the change of the coefficient signs in the first column of the Routh table to determine $\tiny s$ the specific distribution of the roots of the characteristic equation on the plane. The process is as follows:

If the coefficients in the first column of the Routh table are all positive, the roots of the characteristic equations are in $\tiny s$ the left half plane, and the corresponding system is stable.
If the sign of the coefficient in the first column of the Routh table changes, the number of changes is equal to the number of the roots of the characteristic equation on $\tiny s$ the right half plane, and the corresponding system is unstable.

For the specific Rous-Herwitz stability criterion link, please refer to the following link: https://wenku.baidu.com/view/e5fb484f6ad97f192279168884868762cbaebb17.html

Robot dynamics and control study notes (4)-stability analysis of the control system