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A control system is an interconnection that chooses and applies control inputs to influence a controlled plant. A controller may use a reference signal, a measured output and internal state to determine those inputs. The plant, sensing, controller and input mechanism jointly determine the resulting motion. In open-loop control, the commands are chosen without updating them from the measured result; feedback control uses that result to change the commands.

An original dimensionless model shows the distinction. Let the measured output equal the plant state $x$ and let $d$ be a constant control disturbance:The open-loop command $u=r=1$ gives equilibrium $x=1+d$. A controller using both the reference command and measured tracking error can instead setSubstitution gives $\dot{x}=-(1+k)x+(1+k)r+d$. Its equilibrium and transient areWith $k=2$ and $d=0.3$, the equilibrium is1.1 instead of the open-loop value1.3. The controller reacts to the effect of the disturbance through the measured state; it need not measure $d$ separately. With no disturbance, the explicit reference term makes the equilibrium equal to1.

The interconnection matters as much as the controller formula. Measurement error changes the quantity fed back, actuator saturation can prevent delivery of the commanded input, and unmodeled dynamics can change the transient. The assumption that the entire plant state is measured is also special to this example.

Robust control asks how such a closed loop behaves across a family of possible models, distinguishing robust stability from robust performance.^[ref] A complete control-system specification therefore identifies the plant, signals, interconnection, physical limits and response criterion, rather than naming only a controller.

Robust control designs or assesses a controller against a specified family of possible controlled plants, rather than only one nominal model. Control model uncertainty can represent uncertain parameters or unmodeled dynamics. Robust stability asks whether every allowed plant gives a stable closed loop; robust performance also requires a quantitative response criterion throughout the allowed family. Stability alone does not guarantee the required performance.^[ref]

An original scalar example makes the quantifiers explicit. Consider a controlled plant with measured state $x$ and control input $u$:The same feedback gain $k$ must be chosen before knowing $a$. Its solution is $x(t)=x(0)e^{(a-k)t}$. At the nominal value $a=2$, asymptotic stability requires only $k>2$. Choosing $k=2.5$ stabilizes that nominal model but fails at the allowed value $a=3$, where deviations grow as $e^{0.5t}$. Uniform stability over the entire closed interval instead requiresAt $k=3$, the endpoint plant has $\dot{x}=0$ and does not decay, so the strict inequality matters.

Now specify the performance bound $|x(t)|\le |x(0)|e^{-t}$ for every allowed $a$ and all $t\ge 0$. In this example it holds exactly when $k\ge 4$. Thus $k=3.1$ gives robust stability but not the requested rate of decay; the endpoint plant decays only as $e^{-0.1t}$. The performance statement is stronger than the stability statement.

The guarantee remains conditional on its model: exact state measurement, the stated parameter interval and the realizable law $u=-kx$. At $k=4$, an initial state $x(0)=2$ demands an initial input of magnitude8 in the chosen consistent units. An actuator saturation limit of5 invalidates that ideal law for this initial condition. Increasing feedback gain therefore cannot be treated as a cost-free route to robustness. A useful design specifies both the uncertainty family and the physical limits under which its guarantee is meant to apply.

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.