Robust control

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.

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.

Multiplicative uncertainty represents a model error relative to a nominal controlled plant. For a scalar transfer function, one useful form iswhere $G_0$ is nominal, $W$ shapes the permitted relative error across frequency, and $\Delta$ belongs to a specified bounded uncertainty class. Unlike an additive error, the correction is multiplied by the nominal plant. MathWorks's ucover reference distinguishes input multiplicative, output multiplicative and additive forms.^[ref]

For an original static-error example, take $G_0(s)=2/(s+1)$, $W=0.20$ and a real constant $\Delta \in [-1,1]$. The allowed family isThe uncertainty changes gain while preserving the pole in this family. A dynamic $\Delta (s)$ can instead introduce frequency-dependent magnitude and phase changes; its admissible dynamics must be stated separately.

An omitted actuator lag illustrates a dynamic relative error. If $G=G_0/(1+\tau s)$, thenAt $|\omega |\tau =0.1$, its magnitude is about0.0995; at $|\omega |\tau =10$, it is about0.995. A constant10percent relative-error bound would cover the first frequency but fail badly at the second. A shaping function should reflect that discrepancy rather than extrapolate a low-frequency fit.

For a matrix-valued plant, multiplication order matters. Input uncertainty is placed on the right of $G_0$, while output uncertainty is placed on the left; the identity matrices and uncertainty dimensions must match the corresponding signal spaces.^[ref] These forms generally cannot be interchanged when matrix products do not commute.

The scalar ratio $(G-G_0)/G_0$ also needs care near zeros of $G_0$. A tiny absolute discrepancy can be a large relative discrepancy there. Choosing multiplicative uncertainty is therefore a modeling decision about which relative errors and frequency ranges a robust control argument is intended to cover.

Unmodeled dynamics are time-dependent behaviors omitted from the nominal model of a controlled plant or its interconnection. Examples include an omitted actuator lag, an additional flexible mode or a delay. They matter when the neglected behavior acts on time scales relevant to the controller. This differs from changing only the numerical parameters in a fixed equation model.

For an original example, a nominal plant satisfies $\dot{x}=u$, and state feedback control commands $u=-kx$. The nominal prediction is $\dot{x}=-kx$, with decay rate $k>0$. Now suppose the actual applied input follows a first-order actuator:Eliminating $u$ givesWith $k=1$ and $\tau =1$ in chosen consistent units, the exponents are $(-1\pm i\sqrt{3})/2$: the actual response can oscillate while decaying, whereas the nominal model predicts a single exponential. The additional state changes the transient even though the nominal static command remains the same.

The omitted actuator also admits a frequency-domain description. Relative to an ideal actuator with gain1, its transfer function is $1/(1+\tau s)$. The corresponding multiplicative uncertainty isThe relative discrepancy is small when $|\omega |\tau$ is small but approaches1 at high frequency. A low-frequency fit alone therefore need not justify a high-bandwidth design.

Dynamic uncertainty models can describe a family by bounds on its frequency response, with a shaping filter expressing which frequency ranges are poorly known. MathWorks's ultidyn documentation describes this construction for uncertain linear time-invariant systems.^[ref] Such an uncertainty family must be chosen to cover the omitted behavior; its mere presence in a model does not prove that every physical discrepancy has been included.

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.

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 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.