The Concept of Stability and its Necessary Conditions - Control Systems

Stability Conditions

Stability is one of the fundamental requirements of any control system. Informally, a system is called stable if its response remains bounded for every bounded input. More precisely, there are several related definitions of stability used in control theory; each is useful in different contexts. The most commonly used are BIBO stability (bounded-input bounded-output) and internal (or asymptotic) stability. Understanding these definitions and their relationships is essential for analysing linear time-invariant (LTI) systems.

Key definitions

• BIBO stability: An LTI system is BIBO stable if every bounded input produces a bounded output for all time. For rational proper transfer functions, BIBO stability is equivalent to all poles of the transfer function lying strictly in the left half of the complex s-plane.
• Internal stability (asymptotic stability): A system is internally or asymptotically stable if, with zero external input, the natural (zero-input) response due to any finite initial condition decays to zero as time → ∞. For LTI systems described by a polynomial characteristic equation, this requires all roots (poles) to have strictly negative real parts.
• Marginal (neutral) stability: The system neither diverges nor asymptotically decays. This typically occurs when poles lie on the imaginary axis but are simple (non-repeated). In such cases the zero-input response is bounded but does not decay to zero (for example, purely sinusoidal steady oscillations).
• Instability: The system is unstable if any pole has a positive real part (right half-plane) or if there are repeated poles on the imaginary axis, because the response grows without bound or contains unbounded terms.
• Causality and properness: For an LTI rational transfer function to be BIBO stable, the system must be proper (degree of numerator ≤ degree of denominator) and all poles must be in the left half-plane. Non-proper systems or systems with delays/advanced terms require separate care.

Relationship between input, initial conditions and output

It is important to distinguish between the zero-input response (response due only to initial conditions with external input set to zero) and the zero-state response (response due only to external input with zero initial conditions). Internal stability concerns the zero-input response: asymptotic stability requires the zero-input response to decay to zero for any finite initial condition. BIBO stability concerns the zero-state response and requires bounded outputs for bounded inputs.

Simple time-domain examples

Consider the input

x(t) = e^-t

The signal x(t) is bounded and decays to zero as t → ∞. If a system simply gives y(t) = x(t) as output, the output is bounded and decays to zero, showing stable behaviour for this input.

Contrast this with a positive exponential input or response such as y(t) = e^t. This output grows without bound as t increases and therefore demonstrates an unstable response.

Remarks on linearity and zero signals

• For a linear system, if the input is identically zero and the initial conditions are also zero, the output must be identically zero (homogeneity property). However, if initial conditions are non-zero then the output can be non-zero even with zero input; internal stability requires that such zero-input responses decay to zero.
• BIBO stability addresses boundedness of the forced response (zero-state response) due to an external input; internal stability addresses the natural response due to initial conditions. For LTI systems represented by proper rational transfer functions with real coefficients, both concepts are closely related through pole locations.

Stability with respect to the location of poles and zeros

A transfer function of a single-input single-output LTI system is commonly written in factorised form to expose its poles and zeros. For example, consider a general rational transfer function:

In the factorised form the numerator roots are called zeros and the denominator roots are called poles. If the numerator factors yield zeros z1, z2, ..., zm, they are the values of s that make the numerator zero. If the denominator factors yield poles p1, p2, ..., pn, they are the values of s that make the denominator zero and where the transfer function becomes unbounded.

• Zeros and poles may be real or complex conjugate pairs.
• The location of poles in the complex s-plane determines internal stability: all poles must have negative real parts (left half-plane) for asymptotic stability.
• If any pole lies in the right half-plane (positive real part) the system is unstable.
• If poles lie on the imaginary axis, the system is marginally stable if those poles are simple; the system is unstable if there are repeated imaginary-axis poles.

Worked example - identification of poles and zeros

Example: G(s) = (s + 5)(s + 2) / (s + 1)

Equate the numerator to zero to obtain zeros.

(s + 5)(s + 2) = 0

s = -5

s = -2

Thus the zeros are -5 and -2.

Equate the denominator to zero to obtain poles.

(s + 1) = 0

s = -1

Thus the pole is -1.

Because the pole is at s = -1 (left half-plane) and the zeros are also in the left half-plane, the system has its poles in the stable region (this example is a stable, proper transfer function). Pole-zero plots are a convenient visual tool to assess stability quickly.

S-Plane plots and the complex variable s

The complex variable s is written as s = σ + jω, where σ is the real part and ω is the angular frequency. Points on the imaginary axis correspond to σ = 0 and s = jω. When control literature sometimes writes s = jω it is referring specifically to frequency response evaluation along the imaginary axis, but the full stability discussion requires the entire complex s-plane (σ vs ω).

Example: G(s) = 5 (s + 2) / (s² + 2s + 2)

Find zeros and poles.

Numerator: 5 (s + 2) = 0

s = -2

Thus the zero is -2.

Denominator: s² + 2s + 2 = 0

Apply the quadratic formula to find poles.

s = [-2 ± sqrt(2² - 4·1·2)] / 2

s = [-2 ± sqrt(4 - 8)] / 2

s = [-2 ± sqrt(-4)] / 2

s = [-2 ± 2j] / 2

s = -1 ± j

Thus the poles are at -1 + j and -1 - j, which lie in the left half-plane (σ = -1). This example therefore has stable poles.

Summary of s-plane stability rules:

• All poles strictly in the left half-plane ⇒ asymptotically stable (zero-input response decays to zero).
• Poles on the imaginary axis, simple (non-repeated) ⇒ marginal stability (bounded oscillations may remain).
• Any pole in the right half-plane or repeated poles on the imaginary axis ⇒ unstable.
• For proper rational transfer functions with real coefficients, BIBO stability is equivalent to all poles lying in the left half-plane.

Basic concepts of the Routh-Hurwitz criterion

The Routh-Hurwitz criterion is a systematic algebraic test to determine how many roots of a polynomial characteristic equation lie in the right half of the s-plane without explicitly computing the roots. It is widely used to check stability of linear time-invariant systems described by polynomial characteristic equations.

• The Routh-Hurwitz procedure constructs the Routh array from the coefficients of the characteristic polynomial. The number of sign changes in the first column of the array equals the number of roots that lie in the right half-plane.
• The criterion does not provide the actual values or precise locations of the roots; it only indicates how many are in the right half-plane and hence whether the system is stable.
• The criterion does not indicate whether roots are real or complex; it only counts those with positive real parts.
• The method applies to linear systems with a polynomial characteristic equation having real coefficients. If coefficients are not real or the characteristic equation is not polynomial (for example contains exponential terms), the standard Routh procedure does not directly apply.
• Special cases require care: a zero in the first column, or an entire row of zeros, are typical special situations that must be handled using substitution (small ε) or building an auxiliary polynomial (from the even/odd polynomial) and differentiating it to continue the array construction.

Practical notes about using Routh-Hurwitz:

• Form the characteristic polynomial with coefficients ordered by descending powers of s.
• Construct the Routh array rows using these coefficients; evaluate the first column.
• Count sign changes in the first column to find number of right half-plane roots.
• If no sign change occurs and no zero entries produce singularities in the array, all roots are in the left half-plane and the system is stable.

Applications and closing remarks

• Stability analysis is the first step in control system design. Controllers are designed to place closed-loop poles in desired locations (left half-plane) to ensure stability and meet transient/steady-state performance specifications.
• Pole-zero plots, time-domain inspection, frequency-domain methods and algebraic tests such as Routh-Hurwitz complement one another and are chosen according to the available information and design goals.
• For higher-order or more complex systems numerical root-finding or computer-based symbolic tools are commonly used in addition to analytical criteria.

Summary

Stable behaviour in LTI systems is determined principally by the locations of poles in the s-plane: all poles in the left half-plane yield asymptotic stability; poles on the imaginary axis may produce marginal stability; poles in the right half-plane or repeated imaginary-axis poles cause instability. BIBO stability and internal stability are related but distinct concepts; rational, proper LTI systems are BIBO stable when all poles lie in the left half-plane. The Routh-Hurwitz test is a reliable algebraic method to determine the number of right half-plane roots without solving for the roots explicitly.