16.1: General Time-Response Stability Criterion for LTI Systems

For any linear, time-invariant (LTI), single-input-single-output (SISO) physical system, we denote the input as \(u(t)\) [or \(r(t)\), the reference operator setting for controlled systems] and the output as \(x(t)\). For an \(n\)^th order system, in general, the input and output are related by an ODE of the form

\[a_{1} \frac{d^{n} x}{d t^{n}}+a_{2} \frac{d^{n-1} x}{d t^{n-1}}+\ldots+a_{n} \frac{d x}{d t}+a_{n+1} x=b_{1} \frac{d^{m} u}{d t^{m}}+b_{2} \frac{d^{m-1} u}{d t^{m-1}}+\ldots+b_{m} \frac{d u}{d t}+b_{m+1} u\label{eqn:16.1} \]

Symbols \(a_{1}, \ldots, a_{n+1}\) and \(b_{1}, \ldots, b_{m+1}\) are constants (with the numbering system keyed to MATLAB notation), and \(m \leq n\). With use of Equation 2.2.10, especially for initial-condition terms, the Laplace transform of Equation \(\ref{eqn:16.1}\) is

\[\left(a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n} s+a_{n+1}\right) L[x(t)]-a_{1}\left[s^{n-1} x(0)+s^{n-2} \frac{d x}{d t}(0)+\ldots+\frac{d^{n-1} x}{d t^{n-1}}(0)\right]-a_{2}\left[s^{n-2} x(0)+s^{n-3} \frac{d x}{d t}(0)+\ldots+\frac{d^{n-2} x}{d t^{n-2}}(0)\right]-\ldots-a_{n} x(0)=\left(b_{1} s^{m}+b_{2} s^{m-1}+\ldots+b_{m} s+b_{m+1}\right) L[u(t)]\label{eqn:16.2} \]

The initial-condition terms on the left-hand side are algebraically messy; it will suit our purposes to combine together known \(a_{k}\) constants and known initial conditions into constant coefficients denoted as \(I C_{k}\), and to express the initial-condition terms in the following simpler algebraic form, arranged in order of descending powers of \(s\):

\[-\overbrace{a_{1} x(0)}^{=I C_{1}} s^{n-1}-\overbrace{\left[a_{1} \frac{d x}{d t}(0)+a_{2} x(0)\right]}^{\equiv I C_{2}}S^{n-2}-\ldots-\overbrace{\left[a_{1} \frac{d^{n-1} x}{d t^{n-1}}(0)+a_{2} \frac{d^{n-2} x}{d t^{n-2}}(0)+\ldots+a_{n} x(0)\right]}^{\equiv I C_{n}}\equiv-I C_{1} s^{n-1}-I C_{2} s^{n-2}-\ldots-I C_{n-1} s-I C_{n} \nonumber \]

Therefore, we can express Equation \(\ref{eqn:16.2}\) in the algebraically simpler form,

\[\left(a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n} s+a_{n+1}\right) L[x(t)]-I C_{1} s^{n-1}-I C_{2} s^{n-2}-\ldots-I C_{n-1} s-I C_{n}=\left(b_{1} s^{m}+b_{2} s^{m-1}+\ldots+b_{m} s+b_{m+1}\right) L[u(t)]\label{eqn:16.3} \]

The algebraic solution of Equation \(\ref{eqn:16.3}\) for the Laplace transform of the output is:

\[L[x(t)]=\frac{I C_{1} s^{n-1}+I C_{2} s^{n-2}+\ldots+I C_{n-1} s+I C_{n}}{a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n} s+a_{n+1}}+\frac{b_{1} s^{m}+b_{2} s^{m-1}+\ldots+b_{m} s+b_{m+1}}{a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n} s+a_{n+1}} L[u(t)]\label{eqn:16.4} \]

It is appropriate now to examine the nature of transform solution Equation \(\ref{eqn:16.4}\). Let us refer to the first term on the right-hand side as the “IC-response transform,” and to the second term as the “forced-response transform.” Observe first that the coefficient of input transform \(L[u(t)]\) in the forced-response transform is the system transfer function, from Equation 4.6.3,

\[T F(s) \equiv \frac{L[x(t)]_{I C s=0}}{L[u(t)]}=\frac{b_{1} s^{m}+b_{2} s^{m-1}+\ldots+b_{m} s+b_{m+1}}{a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n} s+a_{n+1}}\label{eqn:16.5} \]

If the system is controlled with use of feedback, then Equation \(\ref{eqn:16.5}\) represents the closed-loop transfer function, which we usually denote as \(\operatorname{CLTF}(s)\). It is conventional to express \(T F(s)\) in the following form, with both numerator and denominator polynomials factored:

\[T F(s)=\frac{b_{1} s^{m}+b_{2} s^{m-1}+\ldots+b_{m} s+b_{m+1}}{a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n} s+a_{n+1}} \equiv \frac{\operatorname{Num}(s)}{\operatorname{Den}(s)}=\frac{b_{1}\left(s-z_{1}\right)\left(s-z_{2}\right) \ldots\left(s-z_{m}\right)}{a_{1}\left(s-p_{1}\right)\left(s-p_{2}\right) \ldots\left(s-p_{n}\right)}\label{eqn:16.6} \]

The \(m\) roots \(z_k\) of polynomial equation \(\operatorname{Num}(z)=0\) are called the zeros of the transfer function, since \(T F(s)=0\) if s equals any \(z_k\). The \(n\) roots \(p_k\) of polynomial equation \(\operatorname{Den}(p)=0\) are called the poles of the transfer function, because \(T F(s) \rightarrow \infty\) if \(s\) equals any \(p_k\). [Imagine the complex space surface \(T F(s)\) graphed as a function of coordinates \(\operatorname{Re}(s)\) and \(\operatorname{Im}(s)\): for \(s\) in the vicinity of any \(p_k\), the surface will look something like that of a circus tent near where it is held up by a structural pole; however, whereas the circustent pole has finite length, the mathematical pole is infinitely long.]

Observe next that both right-hand-side terms of Equation \(\ref{eqn:16.4}\) have in their denominators the term \(\operatorname{Den}(s)\) defined in Equation \(\ref{eqn:16.6}\). This term is the key to system stability. The \(n\)^th degree polynomial equation \(\operatorname{Den}(p)=0\) is the general¹ characteristic equation of the system, and its \(n\) roots are the poles \(p_{k}\), \(k=1,2, \ldots, n\). System stability is determined by the signs of the real parts of the poles. The general result is:

If \(\operatorname{Re}\left(p_{k}\right)<0\) for all \(k=1,2, \ldots, n\), then the system is stable. If, on the other hand, there is at least one pole for which \(\operatorname{Re}\left(p_{k}\right)>0\), then the system is unstable.

In order to justify this general stability criterion, we shall continue the analysis of Equation \(\ref{eqn:16.4}\) by deriving the inverse transform conceptually; however, it is appropriate first to state the criterion above, without proof, so that you can see the simple yet very important result of the analysis without being burdened by all the details of the derivation.

First, let us review without proof some basic results from the theory of polynomial equations with real coefficients, a subject covered in most algebra textbooks. The roots of such equations are either real or complex. Moreover, complex roots always appear in conjugate pairs. For example, if one of the roots of \(\operatorname{Den}(p)=0\) is complex, \(p_{k}=\sigma_{k}+j \omega_{k}\), where \(\sigma_{k}\) and \(\omega_{k}\) are real, then one of the other roots is the complex conjugate, \(p_{k+1}=\bar{p}_{k}=\sigma_{k}-j \omega_{k}\).

Returning to Equation \(\ref{eqn:16.4}\), let us evaluate first the IC-response transform. The numerator polynomial is of degree less than \(n\), that of the denominator polynomial, so we can usually expand the entire term into simple partial-fraction form [see the discussion surrounding Equations 2.3.5 and 2.3.6]:

\[\frac{I C_{1} s^{n-1}+I C_{2} s^{n-2}+\ldots+I C_{n-1} s+I C_{n}}{a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n} s+a_{n+1}}=\frac{I C_{1} s^{n-1}+I C_{2} s^{n-2}+\ldots+I C_{n-1} s+I C_{n}}{a_{1}\left(s-p_{1}\right)\left(s-p_{2}\right) \ldots\left(s-p_{n}\right)}=\sum_{k=1}^{n} \frac{C_{k}}{s-p_{k}}\label{eqn:16.7} \]

Right-hand-side constants \(C_{k}\) in Equation \(\ref{eqn:16.7}\) are called residues. Some relevant results from the theory of fractions of polynomials with real coefficients are: if pole \(p_{k}\) is real, then the associated residue \(C_{k}\) also is real; if pole \(p_{k}\) is complex, then the associated residue \(C_{k}\) also is complex, and the residue associated with the complex conjugate pole \(p_{k+1}\) is the conjugate of \(C_{k}\), \(C_{k+1}=\bar{C}_{k}\). It will be useful in derivations that follow to express a complex residue in polar form, \(C_{k} \equiv\left|C_{k}\right| e^{j \angle C_{k}}\), in which \(\left|C_{k}\right|\) is the absolute value (as in the MATLAB function abs), and \(\angle C_{k}\) is the angle (as in the MATLAB function angle). See Section 2.1 for a review of complex numbers and complex arithmetic.

Taking the inverse transform of Equation \(\ref{eqn:16.7}\) to find initial-condition response, we deal, in general, with two “Types” of terms:

1. terms that have real poles and real residues; and
2. pairs of terms that have complex conjugate poles and complex conjugate residues.

The inverse of any Type 1 term is a simple real exponential, from Equation 2.2.6:

\[L^{-1}\left[\frac{C_{k}}{s-p_{k}}\right]=C_{k} e^{p_{k} t}, C_{k} \text { real, and } p_{k} \text { real }\label{eqn:16.8} \]

In Equation \(\ref{eqn:16.8}\), if \(p_{k}<0\), then the term decays exponentially as time increases, which is stable response. Such a term decays eventually to a static equilibrium state of zero; let us call this exponential stability. However, if \(p_{k}>0\), then the term is a permanently growing exponential as time increases, which is unstable response. Finally, if \(p_{k}=0\), then the term is constant in time; such response is bounded and therefore not unstable, which is good for practical purposes; but it is also not so stable that it eventually decays to zero.

Next, let us consider Type 2 terms, as defined above Equation \(\ref{eqn:16.8}\), and derive the inverse transform of a pair of terms that have complex conjugate poles and complex conjugate residues. In the following sequence of operations, we apply: first, Equation 2.2.6; next, identities derived in homework Problems 2.3.1 and 2.4; and, finally, Euler’s equation, Equation 2.1.12:

\[L^{-1}\left[\frac{C_{k}}{s-p_{k}}+\frac{\bar{C}_{k}}{s-\bar{p}_{k}}\right]=C_{k} e^{p_{k} t}+\bar{C}_{k} e^{\bar{p}_{k} t}=C_{k} e^{p_{k} t}+\overline{C_{k} e^{p_{k} t}}=2 \times \operatorname{Re}\left[C_{k} e^{p_{k} t}\right] \nonumber \]

\[=2 \times \operatorname{Re}\left[\left(\left|C_{k}\right| e^{j \angle C_{k}}\right) e^{\left(\sigma_{k}+j \omega_{k}\right) t}\right]=2\left|C_{k}\right| e^{\sigma_{k} t} \times \operatorname{Re}\left[e^{j\left(\omega_{k} t+\angle C_{k}\right)}\right] \nonumber \]

\[=2\left|C_{k}\right| e^{\sigma_{k} t} \cos \left(\omega_{k} t+\angle C_{k}\right)\label{eqn:16.9} \]

Equation \(\ref{eqn:16.9}\)² varies with time as a sinusoid modulated by an exponential envelope. With respect to stability, the critical term is the exponential envelope, \(e^{\sigma_{k} t}\), and its character is determined by the sign of \(\sigma_{k} \equiv \operatorname{Re}\left(p_{k}\right)\). If \(\operatorname{Re}\left(p_{k}\right)<0\), then \(e^{\operatorname{Re}\left(p_{k}\right) t}\) is a decaying exponential, and the total response is a stable positively damped oscillation. However, if \(\operatorname{Re}\left(p_{k}\right)>0\), then \(e^{\operatorname{Re}\left(p_{k}\right) t}\) is a permanently growing exponential, and the total response is an unstable ever-expanding oscillation. Finally, if \(\operatorname{Re}\left(p_{k}\right)=0\), then \(e^{\operatorname{Re}\left(p_{k}\right) t}\) is constant in time, and the total response is a sinusoid of constant magnitude; such response is bounded and therefore not unstable; but it is also not so stable that it eventually decays to a static equilibrium state, i.e., it is not exponentially stable.

Now that we have derived the stability criterion for the IC-response terms of Equation \(\ref{eqn:16.4}\), let us turn our attention to the forced-response transform,

\[L\left[x_{f}(t)\right] \equiv \frac{b_{1} s^{m}+b_{2} s^{m-1}+\ldots+b_{m} s+b_{m+1}}{a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n} s+a_{n+1}} L[u(t)]=\frac{b_{1}\left(s-z_{1}\right)\left(s-z_{2}\right) \ldots\left(s-z_{m}\right)}{a_{1}\left(s-p_{1}\right)\left(s-p_{2}\right) \ldots\left(s-p_{n}\right)} L[u(t)]\label{eqn:16.10} \]

We shall consider explicitly in this derivation only transfer functions for which the following practically common conditions apply: \(m<n\); no pole is repeated; no zero equals any pole. With these restrictions and with another restriction described in the next paragraph, this particular derivation does not apply to all conceivable LTI systems. However, the general derivation is more involved than necessary for our purposes. The stability criterion that we shall derive applies to any LTI system, even though we will not prove it in general. In Section 16.2, a specific, non-trivial system is analyzed in detail, in order to illustrate all of the theory that is developed in this section.

In order to find the conceptual inverse of Equation \(\ref{eqn:16.10}\), let us consider the nature of transform \(L[u(t)]\) for physically common input functions \(u(t)\). In this derivation, we shall consider explicitly only the type of input function whose Laplace transform is a fraction of polynomials in the Laplace variable \(s\), with the numerator polynomial being of lower degree than the denominator polynomial. (However, the same stability criterion that we derive for this special type of input function can be derived for any physically realizable input function.) The following are simple examples of this type of input function, which are defined to be non-zero only for \(t \geq 0\): ideal step \(H(t)\), for which \(L[H(t)]=1 / s\); exponential-unit-impulse \(e^{-t / \tau} / \tau\), for which \(L\left[e^{-t / \tau} / \tau\right]=1 /(\tau s+1)\); exponential-unit-step \(1-e^{-t / 7}\), for which \(L\left[1-e^{-t / \tau}\right]=1 /[s(\tau s+1)]\) (see homework Problem 8.6 for a discussion of the exponential-unit-step and exponential-unit-impulse functions); ramped exponential pulse \(t e^{-t / \tau}\), for which \(L\left[t e^{-t / \tau}\right]=1 /(s+1 / \tau)^{2}\); and sinusoid \(\cos \omega t\), for which \(L\left[t e^{-t / \tau}\right]=1 /(s+1 / \tau)^{2}\). Let us denote the Laplace transform of such input functions in a general form:

\[L[u(t)] \equiv \frac{N_{u}(s)}{D_{u}(s)}=\frac{d_{1} s^{m u}+d_{2} s^{m u-1}+\ldots+d_{m u} s+d_{m u+1}}{c_{1} s^{n u}+c_{2} s^{n u-1}+\ldots+c_{n u} s+c_{n u+1}}=\frac{d_{1}\left(s-w_{1}\right)\left(s-w_{2}\right) \ldots\left(s-w_{m u}\right)}{c_{1}\left(s-q_{1}\right)\left(s-q_{2}\right) \ldots\left(s-q_{n u}\right)}\label{eqn:16.11} \]

Constants \(c_{i}\) and \(d_{i}\) are known coefficients of powers of \(s\), and constants \(q_{i}\) and \(w_{i}\) are, respectively, poles and zeros of the input transform. Keep in mind that we assume \(N_{u}(s)\) in Equation \(\ref{eqn:16.11}\) to have lower degree than \(D_{u}(s)\), \(m u<n u\).

Proceeding with the conceptual forced-response solution, we substitute Equation \(\ref{eqn:16.11}\) into Equation \(\ref{eqn:16.10}\):

\[L\left[x_{f}(t)\right]=\frac{b_{1}\left(s-z_{1}\right)\left(s-z_{2}\right) \ldots\left(s-z_{m}\right)}{a_{1}\left(s-p_{1}\right)\left(s-p_{2}\right) \ldots\left(s-p_{n}\right)} \times \frac{d_{1}\left(s-w_{1}\right)\left(s-w_{2}\right) \ldots\left(s-w_{m u}\right)}{c_{1}\left(s-q_{1}\right)\left(s-q_{2}\right) \ldots\left(s-q_{n u}\right)}\label{eqn:16.12} \]

Forced-response transform Equation \(\ref{eqn:16.12}\) is a fraction of polynomials in \(s\), with numerator polynomial degree \(m+m u\) being less than denominator polynomial degree \(n+n u\). Therefore, we can, in principle, usually expand Equation \(\ref{eqn:16.12}\) into the following form, in which terms that are associated explicitly with the transfer-function poles have simple partial-fraction form, including residues \(C U_{k}\):

\[L\left[x_{f}(t)\right]=\sum_{k=1}^{n} \frac{C U_{k}}{s-p_{k}}+F\left(s ; q_{i}\right)\label{eqn:16.13} \]

The symbol \(F\left(s ; q_{i}\right)\) denotes terms associated explicitly with poles \(q_{i}\) of the input transform. Depending upon the specific nature of \(L[u(t)]\), these terms might or might not have simple partial-fraction form. In any case, the terms in \(F\left(s ; q_{i}\right)\) have the same poles \(q_{i}\) as the input transform, which are always stable poles, so these terms cannot produce instability. Therefore, instability of the forced-response can exist only in the terms \(\sum_{k=1}^{n} C U_{k} /\left(s-p_{k}\right)\), i.e., the terms containing the poles of the system transfer function.

The forced-response terms \(\sum_{k=1}^{n} C U_{k} /\left(s-p_{k}\right)\) are identical in form to the IC-response terms \(\sum_{k=1}^{n} C_{k} /\left(s-p_{k}\right)\) of Equation \(\ref{eqn:16.7}\). Recall that the analysis of stability for ICresponse [Equations \(\ref{eqn:16.8}\) and \(\ref{eqn:16.9}\) and the associated discussion] was based upon that form. Therefore, the analysis of stability for forced-response is identical to that for ICresponse, so we need not repeat it. However, it is worth repeating, for emphasis, the general stability criterion for LTI systems that has been derived (albeit, not generally) here:

For an n^th order LTI system, the poles of the transfer function are denoted as \(p_{k}\), \(k=1,2, \ldots, n\). If \(\operatorname{Re}\left(p_{k}\right)<0\) for all \(k=1,2, \ldots, n\), then the system is stable. If, on the other hand, there is at least one pole for which \(\operatorname{Re}\left(p_{k}\right)>0\), then the system is unstable.

Observe from the derivation that the stability or instability of an LTI system does not depend upon ICs or forcing—it is an intrinsic property of the system. However, by referring to general transform solution Equation \(\ref{eqn:16.4}\), one might make the mathematical argument that if all initial conditions were zero, and if forcing input \(u(t)\) also were zero, then the solution would be \(x(t)=0\), and therefore even a system that has an unstable pole, \(\operatorname{Re}\left(p_{k}\right)>0\), would be stable if there were no stimulus. In physical reality, though, there is always some perturbation or excitation, no matter how small, acting upon a system; and even the tiniest IC or forcing input, which would be negligible for a stable system, will always be enough to provoke the instability of an unstable LTI system.

¹The characteristic equation of the free-vibration problem described in Chapter 12 is a specific form, applicable only to undamped free vibration, of this general characteristic equation.

²When applying Equation \(\ref{eqn:16.9}\), you should assign the title of \(p_{k}\) to the one root (of the complex conjugate pair of roots) for which \(\operatorname{Im}\left(p_{k}\right) \equiv \omega_{k}>0\), in order to define the oscillatory frequency as being positive. Then you should assign the title of \(C_{k}\) to the residue that is associated with the root for which \(\operatorname{Im}\left(p_{k}\right)>0\).