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Control model uncertainty specifies how the behavior of a controlled plant may differ from a nominal model. A robust control claim is made over this allowed family. The description includes which parameters may change, their ranges, and which occurrences must represent the same physical quantity. It can also include unmodeled dynamics, such as an omitted actuator response, that cannot be represented by varying one constant coefficient.

Parametric model uncertainty retains a chosen equation form while allowing its parameters to vary. For an original example, a damped mass obeysHere $v$ is velocity and $u$ the applied-force control input. In the form $\dot{v}=av+bu$, the coefficients areTheir numerical ranges in consistent SI units are approximately $a\in [-2.222,-1.818]$ and $b\in [0.909,1.111]$, but they are not independent. The combination $a=-2.222$, $b=0.909$ does not correspond to one allowed mass. Replacing the physically linked family by the full rectangle of independent coefficient values introduces additional possible models.

This dependency matters when claiming robust stability or robust performance: enlarging the family may produce a more conservative test, while accidentally narrowing it can remove a genuinely possible failure. MathWorks's uncertain-parameter representation tracks the identity of named uncertain elements when they recur in a model.^[ref]

A bounded uncertainty set does not by itself assign probabilities. Saying $m\in [0.9,1.1]$ does not say that the endpoints are unlikely or that all masses are equally likely. It states what the guarantee must cover. Conversely, fitting the nominal mass from data does not prove that this interval covers every relevant operating condition.

The assumed time behavior is also part of the specification. An unknown but fixed mass and an arbitrarily time-varying coefficient are different model families. Likewise, uncertain gain does not automatically cover an unmodeled delay. A useful uncertainty description connects allowed mathematical variations to the physical mechanisms and measurements that justify them.

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.

An induced signal gain is the smallest norm-based amplification bound obtained by taking the worst case over admissible inputs. For a zero-initial-state operator $T$ satisfying $T0=0$,For a linear map this is an operator norm between the specified normed spaces. Axler's definition takes the supremum of output norm over inputs of norm at most1, which is equivalent by linear scaling.^[ref]

The norms are part of the definition. An original finite-dimensional example usesWith the same maximum-component norm at input and output, $\Vert Mu{\Vert }_{\infty }\le 2\Vert u{\Vert }_{\infty }$, and $u=(1,1)$ attains the ratio2. With the Euclidean norm, the Cauchy-Schwarz inequality gives $|u_1+u_2|\le \sqrt{2}\Vert u{\Vert }_2$, with equality at $u=(1,1)$. With the sum-of-absolute-components norm, $|u_1+u_2|\le |u_1|+|u_2|$, and $u=(1,0)$ attains ratio1. The same matrix therefore has induced gains2, $\sqrt{2}$ and1 under these three conventions.

For a time-dependent signal, the choice may additionally compare peak amplitudes or integrated squared amplitudes. The example in input-output gain derives a sharp peak-amplitude bound for a scalar dynamic system. A large response to one trial input supplies a lower bound on induced gain; proving a universal upper bound requires controlling every admissible input.

Compatible induced bounds compose. If $\Vert T_{1}u\Vert \le {\gamma }_1\Vert u\Vert$ and $\Vert T_{2}v\Vert \le {\gamma }_2\Vert v\Vert$, thenThe intermediate norms must be the same. This inequality bounds a cascade without requiring the inputs that maximize the individual blocks to be simultaneously realizable, so the product need not be sharp. In robust performance, both the signal norms and the allowed control model uncertainty must be stated before such a bound can be interpreted.

An input-output gain bounds how large a system's output can be relative to its input, after the input and output measures have been specified. For a zero-initial-state operator $T$ and chosen signal norms, its induced signal gain isA finite gain gives $\Vert y{\Vert }_{\mathrm{out}}\le \gamma \Vert u{\Vert }_{\mathrm{in}}$. The supremum is over the admissible input class, not merely a collection of trial inputs. Initial-state response must be treated separately.

For an original scalar linear time-invariant system, letUsing the time-domain L-infinity norm, the convolution solution obeysA constant input of suitable sign approaches the limiting ratio $|b|/a$, so this bound is sharp. At $a=2$ and $b=3$, the peak-amplitude gain is1.5: an input bounded by2 gives output bounded by3 from zero initial state, and the constant input approaches that value.

The signal norms change the problem. For the same stable scalar system, the peak frequency response magnitude isIt coincides numerically with the preceding gain in this example. For stable linear systems, this frequency-domain quantity is the energy-signal induced gain associated with the time-domain L2 norm; it should not generally be identified with the peak-amplitude induced gain. MathWorks's linear-model norm documentation distinguishes this peak frequency-response quantity from its separate impulse-response norm.^[ref]

The units matter as well: a gain from force input to displacement output has physical units, whereas identical normalized signal units can produce a dimensionless gain. In robust performance, the bound must hold across the allowed control model uncertainty. A nominal gain calculation alone supplies no such universal guarantee.

Parametric model uncertainty allows specified parameters to vary while retaining a chosen mathematical model structure. It records a nominal value and an allowed set, such as an interval, for each uncertain quantity. Repeated occurrences of the same physical parameter must remain linked. MathWorks's uncertain-real-parameter model explicitly represents nominal values, ranges and named repeated parameters.^[ref]

An original RC circuit example uses the first-order transfer functionAssume every resistance/capacitance pair in those positive intervals is allowed. Since their product increases with either factor, the time constant has the exact rangeThe nominal product is1millisecond, so separate symmetric10percent component ranges give an asymmetric time-constant range of minus19percent to plus21percent. The angular frequency at which this model's magnitude falls to $1/\sqrt{2}$ is ${\omega }_c=1/\tau$, giving approximately $826.4$ to $1234.6\mathrm{rad}{\mathrm{s}}^{-1}$. Reciprocal conversion reverses the endpoints.

This family changes response speed while preserving the modeled DC gain $H(0)=1$. A guarantee of correct DC gain therefore does not establish a particular transient or bandwidth. Conversely, an extra pole from unmodeled dynamics is absent from every member of this first-order family, regardless of how widely $R$ and $C$ are allowed to vary.

Parameter dependencies can restrict the family further. If the same uncertain resistor appears in several coefficients, independently varying its appearances changes the physical model. The mass example in control model uncertainty derives one such coefficient relation. Parameter bounds are also not a probability distribution: an interval describes what must be covered, while probability assignments require additional information.

Robust stability and robust performance can then be assessed over this specified family. The result applies to the model structure and bounds actually supplied, not automatically to aging, temperature changes or other effects outside them.

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.

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.