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Actuator lag is delayed adjustment of an actuator's actual output to its commanded value. A simple dynamic model iswhere $u_{\mathrm{cmd}}$ is the command, $u$ the delivered quantity, $K$ the steady-state conversion and $\tau$ the time constant. A first-order actuator model with an estimated time constant appears in MathWorks's aircraft-parameter estimation example.^[ref]

For a step command of amplitude $U$ from zero initial output,The response reaches about63.2percent of its final value at one time constant and95.0percent at three time constants. In an original example with $K=1$, $U=10$ output units and $\tau =0.20\mathrm{s}$, the delivered value at0.20seconds is6.32, and its95percent rise takes about0.599seconds. The command itself has already reached10 at time zero.

The same model's frequency response is $K/(1+i\omega \tau )$. At $\omega \tau =1$, its magnitude is $|K|/\sqrt{2}$ and its phase lag is45degrees for positive $K$. Thus an actuator that is accurate for a constant command can still attenuate and phase-shift a rapidly changing command.

A direct identification experiment records the command and the actual output on a shared time base, applies a small step around an operating point, and estimates the final change and time to63.2percent. This is an explanatory procedure for the model, not a claim about a particular measured actuator. Sensor filtering, delay and an unknown initial state can alter the fitted response and must be included when relevant.

In a control system, treating $u=u_{\mathrm{cmd}}$ discards this state. The example in unmodeled dynamics shows how doing so changes closed-loop motion from a single exponential to a damped oscillation. Physical actuator saturation and rate limits are additional effects; the linear first-order lag equation does not represent them automatically.

Robust performance requires a control system to meet a specified quantitative response criterion throughout its allowed control model uncertainty. It includes the relevant stability requirement, but adds constraints such as disturbance attenuation, tracking accuracy or decay rate. MathWorks's robust-analysis example explicitly separates stability of the uncertain loop from maintaining its prescribed performance level.^[ref]

An original example uses a stable scalar family with a bounded control disturbance $d$:The solution and a direct bound areThe constant disturbance $d(t)=D$ approaches $D/a$, so the worst-case input-output gain for this peak-amplitude criterion is $1/a$. At nominal $a=1.5$, the limiting response is $2D/3$, satisfying a proposed bound $|x(t)|\le 0.75D$. The allowed endpoint $a=1$ instead approaches $D$, violating that requirement despite remaining stable.

Now add state feedback control $u=-kx$ to the right-hand side, with $k\ge 0$. The corresponding worst-case gain becomesThe0.75gain requirement therefore holds for every allowed $a$ exactly when $k\ge 1/3$. This is a statement about a stipulated peak-amplitude bound, not a numerical output from the software example.

The criterion must include its input and initial-state assumptions. A nonzero $x(0)$ contributes $e^{-(a+k)t}x(0)$, and actuator saturation can prevent realization of $u=-kx$. Changing the disturbance norm, measured output or uncertainty family changes the performance problem. Robust performance is therefore a precise guarantee about a stated comparison, rather than a claim that a controller performs well under unspecified conditions.

Robust stability means that a closed-loop control system remains stable for every allowed realization of its control model uncertainty. The intended notion of stability and the uncertainty family must be stated. A finite list of stable simulations does not by itself establish the universal statement. Robust-analysis software reports margins relative to the uncertainty encoded in the model, rather than relative to every imaginable model error.^[ref]

For an original continuous-time linear family, considerA common Lyapunov function is $V(x)=x_1^2+x_2^2$. Along every member of the family,The uncertain cross terms cancel because the same $\theta$ occurs with opposite signs. Consequently,This proves exponential stability uniformly over the whole interval using the Euclidean norm, without sampling possible values of $\theta$. The inequality even remains valid for a bounded measurable $\theta (t)$ in this particular differential equation, because its derivation is pointwise and does not differentiate $\theta$.

The preserved parameter dependence is essential to this certificate. If the off-diagonal entries are instead independent values both equal to10, the matrix $\left(\begin{array}{cc}-1& 10\\ 10& -2\end{array}\right)$ has determinant $-98$ and eigenvalues of opposite signs. That different uncertainty family contains an unstable system.

Robust stability addresses growth and decay of internal motion. Robust performance additionally imposes a response bound, such as disturbance attenuation. A system can satisfy the first requirement while allowing an unacceptably large response to a bounded control disturbance. Conversely, a claimed performance guarantee is meaningless outside the uncertainty family for which its stability assumptions hold.