2.3: System Stability

Stability is a desired characteristic of any dynamic system; it refers to the system being well behaved and in control under various operating conditions. Stability may be categorized in multiple ways, some of which are discussed below.

Bounded-Input Bounded-Output Stability (BIBO)

The bounded-input bounded-output (BIBO) stability implies that for every bounded input, \(u(t)\, :\, \, |u(t)|<M_{1} <\infty ,\, \, \,\) the output of the system stays bounded, that is, \(y(t)\, :\, \, |y(t)|<M_{2} <\infty\).

The output of the system in time-domain is given in terms of the convolution integral: \[y(t)=\int_0^\infty g(t-\tau) u(\tau) d\tau \nonumber \]

where \(g(t)\) is the impulse response of the system. Hence, a necessary condition for BIBO stability is that the impulse response dies out with time, that is, \( \lim_{t\to \infty } g(t)=0\).

The impulse response contains the modes of system natural response and is given as: \[g (t)=\sum _{i=1}^{n} A_{i} e^{p_{i} t} \nonumber \] where \(p_i\) is a pole of the system transfer function. Hence, a necessary condition for BIBO stability is: \({\rm R}e\left[p_{i} \right]<0\).

Physically, the condition \({\rm R}e[p_{i} ]<0\) implies the presence of damping in the system, where the damping terms in the transfer function indicate dissipation of residual energy with time.

Marginal Stability

The imaginary axis on the complex plane serves as the stability boundary. A system with poles in the open left-half plane (OLHP) is stable.

If the system transfer function has simple poles that are located on the imaginary axis, it is termed as marginally stable. The impulse response of such systems does not go to zero as \(t\to\infty\), but stays bounded in the steady-state.

As an example, a simple harmonic oscillator is described by the ODE: \(\ddot{y}+\omega _{n}^{2} y=0\), where \(\omega _{n}\) represents the natural frequency and the system has no damping. The oscillator transfer function, \(G(s)=\frac{1}{s^{2} +\omega _{n}^{2} }\) has simple poles \(\left(p_{1,2}=\pm j{\omega }_n\right)\) on the \(j\omega\)-axis.

The natural response of the simple harmonic oscillator contains the response mode: \(e^{j\omega_n t}=\cos\omega_n t+j\sin\omega_n t\). Its impulse response displays persisting oscillations at the natural frequency.

Internal Stability

The notion of internal stability requires that all signals within a control system remain bounded for every bounded input. It further implies that all relevant transfer functions between input–output pairs in a feedback control system are BIBO stable.

Internal stability is a stronger notion than BIBO stability. It is so because the internal modes of system response may include those modes not be reflected in the input-output transfer function.

In the case of linear system models involving feedback, the internal stability requirements are met if the closed-loop characteristic polynomial is stable and any pole-zero cancelations appearing in the loop gains are restricted to the OLHP.

In particular, for a single-input single-output (SISO) feedback control system, the loop gain includes the product of the plant and the controller transfer functions.

Suppose the plant transfer function is: \(G(s)=\frac{1}{s+1}\), and the controller is given as: \(K(s)=K\left(\frac{s+1}{s+10}\right)\); then, \(K(s)G(s)=\frac{K(s+1)}{(s+1)(s+10)}\), which includes an OLHP pole-zero cancelation. However, the closed-loop characteristic polynomial: \(\Delta(s)=s+10+K\) has stable roots for \(K>-10\). Hence, the closed-loop system is internally stable for \(K>-10\).

Asymptotic Stability

For a general nonlinear system model, \(\dot{x}\left(t\right)=f\left(x,u\right)\), stability refers to the stability of an equilibrium point \(\left(x_e,u_e\right)\) defined by: \(f\left(x_e,u_e\right)=0\).

In particular, the equilibrium point is said to be stable if a system trajectory, \(x\left(t\right)\), that starts in the vicinity of \(x_e\) stays close to \(x_e\). The equilibrium point is said to be asymptotically stable if a system trajectory \(x\left(t\right)\) that starts in the vicinity of \(x_e\) converges to \(x_e\).

For linear system models, defined by: \(\dot x=Ax(t)+Bu(t)\), the origin \(x_e=0\) serves as an equilibrium point. In such cases, the asymptotic stability requires that the poles of the system transfer function (equivalently, the eigenvalues of the system matrix) lie in the open left-half plane, or \({\rm R}e\left[p_{i} \right]<0\), and there are no RHP pole-zero cancelations.