The tool DC-03 detects the stability of linear time-invariant systems by the Routh-Hurwitz stability criterion. Analytical stability conditions are provided for systems with up to order four.

In addition, a tutorial on how to use the tool in computation is provided and a subject review on fundamental theories and useful formulas is presented.

## Tutorial

Example: A third-order system

The input in the tool is After clicking "Run", the results will be shown as follows in another page.

The third order system:    s3 + (2.0) s2 + (1.0) s + (1.0) = 0

Stability by Routh Criterion

Necessary Condition (All ai > 0) met.

Routh Array:

 s3 | 1 1 s2 | 2 1 s1 | 0.5 0 s0 | 1 0

Sufficient Condition met.

Conclusion: The system is stable.

## Subject Review

### 1. Stability of Linear Time-Invariant Systems

A stable system is defined as one with a bounded response to a bounded input. For a linear time-invariant system represented by transfer function with coefficients a’s and b’s being constants, its stability depends on the locations of the transfer function poles, which are the roots of the characteristic equation Let the system poles (the roots of Eq. (2)) be p1, p2,..., pn. There are three stability cases: stability (asymptotic stability), marginal stability, and instability, which are detailed as follows.

Case 1. The system is stable if all the poles have negative real parts, See Fig. 1 for an example, where the crosses represent pole locations in the complex plane. Figure 1. Pole locations of a stable system

Case 2. A system is marginally stable or neutrally stable if the following two conditions are satisfied:

(a) All the poles have non-negative real parts, (b) No repeated poles are on the imaginary axis of the complex plane (including the origin).

See Fig. 2 for three examples. Figure 2. Pole locations of marginally stable systems

Case 3. A system is unstable if one of the following two conditions is true:

(a) At least one pole has positive real part.

(b) There are repeated poles are on the imaginary axis of the complex plane (including the origin).

See Fig. 3 for three examples. Figure 3. Pole locations of unstable systems

### 2. The Routh-Hurwitz Stability Criterion

Routh-Hurwitz stability criterion provides necessary and sufficient conditions for the stability of linear time-invariant systems, which do not require the determination of transfer function poles. To present the stability criterion, consider a system with the characteristic equation (2).

Necessary Condition. All the coefficients in Eq. (2) are positive; that is In other words, if any coefficient of Eq. (2) is negative or zero, the system described by Eq. (1) cannot be stable. In this case, the system is either marginally stable or unstable.

Sufficient Condition. The sufficient condition is deduced from an array shown in Fig. 4, which is often called the Routh array. Figure 4. The Routh array

The first two rows of the array are constructed by properly ordering the coefficients of the characteristic equation, as shown in Fig. 5. Figure 5. Selection of the coefficients of the characteristic equation

The remaining rows of the array are obtained by where a zero is filled in when two adjacent rows do not have the same number of elements.

According to the Routh-Hurwitz criterion, the number of unstable poles (i.e., poles with positive real parts) is the number of sign changes in the first column of the Routh array. It follows that the sufficient condition for the system to be stable is Note: During construction of the Routh array, the first element of a row may be zero. In this case, the system cannot be stable; it is either unstable or marginally stable. Details on how to construct the array and how to examine system stability status can be found from the references listed at the end of this note.

### 3. Stability Conditions for Some Systems

The Routh-Hurwitz stability criterion is applied to several systems and the stability conditions are given below.

(a) First-order system (n = 1)

The characteristic equation of a first-order system is The necessary condition and sufficient condition are the same: which, of course, is trivial.

(b)Second-order system (n = 2)

The characteristic equation of a second-order system is The necessary and sufficient conditions are the same: (c) Third-order system (n = 3)

The characteristic equation of a third-order system is The necessary conditions are The sufficient conditions are As can be seen, the necessary conditions and sufficient conditions for a third-order system are not the same.

(d) Fourth-order system (n = 4)

The characteristic equation of a fourth-order system is The necessary conditions are The sufficient conditions are Again, the necessary conditions and sufficient conditions are not the same.

Example. Consider the feedback control system in Fig. 6. Figure 6. A feedback control system

The characteristic equation of the control system is According to the conditions (13), the control system is stable if ### References

1. R.C. Dorf and R.H. Bishop, 2010, Modern Control Systems, 12th edition, Prentice Hall.

2. K. Ogata, 2009, Modern Control Engineering, 5th Edition, Prentice Hall.

3. N.S. Nise, 2010, Control Systems Engineering, 5th Edition, Wiley.