11. Robust Control and Uncertainty Management

Introduction and context

Classical control design often begins with a nominal mathematical model of a plant. The model may be obtained from physical laws, identification experiments, simulation, or a combination of these sources. A controller is then designed so that the nominal closed loop is stable and meets requirements such as tracking accuracy, disturbance rejection, bandwidth, and transient response. In an actual engineering system, however, the nominal model is never exact. Component tolerances, neglected dynamics, changing operating conditions, sensor noise, actuator limitations, ageing, nonlinearities, and environmental variation all create discrepancies between the model and the plant.

Robust control addresses the question that nominal design leaves unanswered: will the closed-loop system remain stable and perform acceptably for every plant belonging to a specified family of possible plants? This shifts attention from one model to a set of models. The objective is not merely good nominal performance, but an acceptable compromise among performance, stability margin, model fidelity, control effort, and implementation complexity.

The subject connects naturally with frequency-response methods, state-space control, feedback stability, and optimal control. Sensitivity functions quantify how disturbances and modelling errors affect the closed loop. Small-gain reasoning gives a powerful sufficient condition for robust stability. Structured singular value analysis extends this reasoning when uncertainty has known structure. H2 control emphasizes average or energy-based performance, whereas H-infinity control emphasizes worst-case performance. Loop shaping provides an engineering route for placing bandwidth and robustness requirements in a transparent frequency-domain form.

Learning objectives

After studying this chapter, the reader should be able to:

1. distinguish parametric, dynamic, additive, multiplicative, and structured uncertainty;

2. represent uncertain plants using suitable mathematical models;

3. interpret sensitivity and complementary sensitivity functions in robustness analysis;

4. apply the small-gain theorem to assess robust stability;

5. explain the role of the structured singular value and its upper and lower bounds;

6. differentiate H2, H-infinity, and mixed-sensitivity design objectives;

7. use loop-shaping ideas to balance tracking, disturbance rejection, noise attenuation, and robustness;

8. recognize practical limitations involving nonlinearities, saturation, unmodelled dynamics, and conservatism;

9. select an appropriate robust-control method for an engineering application; and

10. evaluate robust designs through simulation, worst-case analysis, and implementation-oriented validation.

Why uncertainty must be modelled explicitly

Suppose a nominal plant is represented by

\[

P_0(s)=\frac{1}{s(s+1)}.

\]

A controller designed for this model may fail if the true plant has an additional flexible mode, a gain variation, a time delay, or a pole that moves close to the imaginary axis. Merely simulating several alternative plants is useful but does not constitute a general guarantee unless the alternatives adequately represent the complete uncertainty set.

Uncertainty modelling therefore has two purposes. First, it describes what is not known about the plant. Second, it provides a mathematical interface through which stability and performance guarantees can be derived. A model that is too narrow gives false confidence; one that is excessively broad may lead to an unnecessarily conservative controller.

Parametric uncertainty

Parametric uncertainty occurs when the form of the model is known but one or more parameters are uncertain. For example,

\[

P(s,q)=\frac{k}{ms^2+bs+k_s},

\]

may contain uncertain mass \(m\), damping \(b\), stiffness \(k_s\), and actuator gain \(k\). A common representation is

\[

q_i=q_{i0}(1+\delta_i), \qquad |\delta_i|\leq 1,

\]

where \(q_{i0}\) is the nominal value and the bound describes the allowed deviation. Parametric uncertainty is valuable when physical parameters have clear meanings and reliable bounds. It can, however, become difficult to analyze when parameters enter the transfer function nonlinearly or when several parameters vary dependently.

Unstructured dynamic uncertainty

Unstructured uncertainty represents unknown dynamics without specifying their internal physical form. Two widely used descriptions are additive and multiplicative uncertainty. In additive form,

\[

P(s)=P_0(s)+W_a(s)\Delta_a(s), \qquad \|\Delta_a\|_\infty\leq 1.

\]

Here, \(W_a(s)\) weights the magnitude and frequency distribution of the modelling error. In multiplicative output uncertainty,

\[

P(s)=P_0(s)[I+W_m(s)\Delta_m(s)], \qquad \|\Delta_m\|_\infty\leq 1.

\]

Multiplicative uncertainty is particularly natural when the relative error \((P-P_0)/P_0\) is known. It generally becomes more important at high frequencies, where neglected modes and actuator or sensor dynamics are common. The weighting function must be based on evidence such as identification data, physical knowledge, frequency-response comparisons, and conservative engineering judgement.

Structured uncertainty

Uncertainty is structured when its form is known. For example, a model may contain two independent real parameter variations and one repeated complex dynamic block:

\[

\Delta=\operatorname{diag}(\delta_1 I_{r_1},\delta_2 I_{r_2},\Delta_3),

\]

where real scalars and full complex blocks have different interpretations. Treating all uncertainty as one unstructured block ignores this information and can produce unnecessarily conservative results. Structured singular value analysis is designed to exploit such information.

Sensitivity as the language of robustness

For a plant \(P\) and controller \(K\) in a standard negative-feedback loop, define the loop transfer function

\[

L=PK.

\]

The sensitivity and complementary sensitivity functions are

\[

S=(I+L)^{-1}, \qquad T=L(I+L)^{-1}=I-S.

\]

For scalar systems, these reduce to ordinary transfer functions. The sensitivity function describes the response from reference-independent disturbances entering at suitable points and measures the effect of plant perturbations near the loop. Low \(|S|\) at low frequencies generally implies good tracking and rejection of plant-input disturbances. The complementary sensitivity \(T\) describes transmission of measurement noise and high-frequency closed-loop response. Low \(|T|\) at high frequencies is therefore usually desirable.

The identity \(S+T=I\) expresses a fundamental trade-off. Improving disturbance rejection by reducing \(S\) over a broad frequency range generally increases \(T\) somewhere else. For stable, proper systems, integral constraints such as the Bode sensitivity relationship show that sensitivity reduction cannot be achieved uniformly at all frequencies. The resulting peak, often written

\[

M_s=\|S\|_\infty,

\]

is a central robustness indicator. A large \(M_s\) implies a pronounced sensitivity peak and typically reduced gain and phase margins. Similarly,

\[

M_t=\|T\|_\infty

\]

indicates the maximum amplification of measurement noise or resonant response.

For multiplicative output uncertainty, robust stability is related to

\[

\|W_mT\|_\infty<1.

\]

Thus the controller must reduce complementary sensitivity where the relative plant uncertainty is large. This explains why aggressive high-frequency loop gain is dangerous: although it may improve tracking over part of the spectrum, it can amplify neglected dynamics and violate the robust-stability condition.

Small-gain reasoning and robust stability

The small-gain theorem states, in one common form, that the feedback interconnection of two stable systems \(M\) and \(\Delta\) is robustly stable for every stable uncertainty satisfying \(\|\Delta\|_\infty\leq 1\) if

\[

\|M\|_\infty<1.

\]

Consider multiplicative uncertainty with a stable nominal closed loop. The uncertain feedback path can be represented so that the relevant transfer matrix is \(M=W_mT\). A sufficient robust-stability condition is therefore

\[

\|W_mT\|_\infty<1.

\]

For additive uncertainty, the analogous condition commonly takes the form

\[

\|W_aKS\|_\infty<1,

\]

assuming the interconnection and weighting definitions are consistent. The interpretation is direct: the closed-loop transfer through which the uncertainty acts must be small enough that the uncertainty cannot reinforce itself into instability.

The condition is powerful but can be conservative. It treats an uncertainty block as capable of selecting its worst phase and frequency behaviour independently. Real physical uncertainties may have restrictions that this representation does not capture. Nevertheless, small-gain analysis is attractive because it provides a clear certificate and applies to nonlinear or time-varying uncertainties under suitable induced-norm assumptions.

Robust performance requires more than robust stability. If an exogenous input \(w\) produces regulated output \(z\), one may require

\[

\|T_{zw}\|_\infty<\gamma.

\]

The value \(\gamma\) is a worst-case induced-gain bound. Unlike nominal frequency-response testing, this requirement covers all disturbances in the selected signal class and all uncertainties admitted by the model, subject to the validity of the assumptions.

Structured singular value analysis

The structured singular value, denoted by \(\mu\), quantifies the smallest structured uncertainty capable of making a matrix singular. For a complex matrix \(M\) and an uncertainty set \(\Delta\), one definition is

\[

\mu_\Delta(M)=\frac{1}{\min\{\bar\sigma(\Delta):\Delta\in\Delta,\ \det(I-M\Delta)=0\}},

\]

with \(\mu_\Delta(M)=0\) if no destabilizing uncertainty exists. Here \(\bar\sigma\) denotes maximum singular value. A large value of \(\mu\) indicates that a relatively small structured perturbation can cause loss of stability or performance.

For a normalized uncertainty set, robust stability at a frequency is guaranteed when

\[

\mu_\Delta(M(j\omega))<1.

\]

A robust-stability margin can be interpreted approximately as the reciprocal of the peak structured singular value:

\[

\text{margin}\approx \frac{1}{\sup_\omega \mu_\Delta(M(j\omega))}.

\]

Exact computation of \(\mu\) is generally difficult. Practical algorithms calculate upper and lower bounds. The lower bound demonstrates that a destabilizing perturbation of a certain size exists; the upper bound provides a sufficient certificate that no smaller perturbation can destabilize the interconnection. A wide gap between bounds indicates that the analysis is inconclusive rather than that the system is necessarily unsafe.

D-scalings are commonly used to obtain computable upper bounds. They scale uncertainty channels to reduce conservatism in a structured small-gain test. In a design workflow, one often uses \(\mu\)-analysis after an initial H-infinity or loop-shaping design, then modifies weighting functions, controller order, or architecture according to the frequencies and channels responsible for the worst-case value.

H2 and H-infinity design perspectives

H2 control minimizes an energy-like measure of closed-loop response. For a stable transfer matrix \(G\), the H2 norm is related to the integral of squared frequency response and, in state-space form, to the steady-state output energy produced by white-noise-like disturbances. H2 design is therefore appropriate when statistical disturbance assumptions are credible and average performance is more important than the absolute worst case.

H-infinity control minimizes the peak singular value:

\[

\|G\|_\infty=\sup_\omega \bar\sigma[G(j\omega)].

\]

It is naturally suited to worst-case disturbance attenuation and robust stability. A standard formulation seeks a controller such that

\[

\|T_{zw}\|_\infty<\gamma,

\]

while stabilizing the generalized plant. The smallest achievable \(\gamma\) is an optimal value; a slightly larger, feasible value is often selected to obtain a lower-order or more implementable controller.

The distinction should not be reduced to “H2 is nominal and H-infinity is robust.” H2 controllers can include robustness considerations through the generalized plant and uncertainty weights, while H-infinity controllers can be designed for nominal performance. The essential difference is the performance metric: integrated energy versus worst-case peak gain.

Mixed-sensitivity formulation

A widely used robust-control formulation minimizes the weighted sensitivity channels simultaneously:

\[

\left\|\begin{bmatrix}W_1S\\W_2KS\\W_3T\end{bmatrix}\right\|_\infty<1.

\]

The first weight, \(W_1\), specifies desired tracking and low-frequency disturbance rejection. Since \(S\) should be small where performance is demanded, \(W_1\) is made large in that frequency range. The second weight, \(W_2\), penalizes control action and protects actuators. The third, \(W_3\), penalizes high-frequency complementary sensitivity and may also represent multiplicative uncertainty.

A typical low-frequency performance weight has large magnitude at low frequencies and decreases with frequency. A control-effort weight may be selected according to actuator ratings, while an uncertainty weight increases at frequencies where the nominal model is unreliable. Weight selection is not a purely mathematical exercise: unrealistic weights can make the problem infeasible, whereas weak weights produce a controller that satisfies the optimization but not the engineering intent.

Generalized-plant construction allows references, disturbances, measurement noise, plant uncertainty, and regulated outputs to be connected in one model. The resulting controller is then evaluated not only by the achieved norm but also by time-domain responses, actuator signals, uncertainty margins, and sensitivity plots.

Loop shaping: an engineering route to robustness

Loop shaping begins with the desired open-loop transfer function rather than directly solving an optimization problem. The designer chooses a controller so that the loop gain has high magnitude at low frequencies for tracking and disturbance rejection, crosses unity with adequate phase margin, and rolls off sufficiently at high frequencies to avoid noise amplification and unmodelled dynamics.

A useful design sequence is:

1. establish the frequency range in which tracking and disturbance rejection are required;

2. identify frequencies at which uncertainty, resonances, delays, and sensor noise become significant;

3. select a target loop shape with suitable low-frequency gain, crossover frequency, and roll-off;

4. use lead, lag, notch, proportional-integral, or state-space compensation to approach the target;

5. evaluate \(S\), \(T\), control effort, disk margins, and uncertainty-weighted gains; and

6. iterate while checking saturation, sampling, quantization, and unmodelled modes.

Normalized coprime-factor loop shaping offers a more formal robust-stability margin than classical gain and phase margins. In practice, the central principle remains the same: bandwidth should be high enough to meet performance requirements but low enough to remain separated from poorly modelled dynamics and actuator limitations.

An illustrative case: robust speed control of an electric drive

Background

Consider an illustrative permanent-magnet motor drive required to regulate shaft speed over a range of loads. A nominal linear model is obtained around one operating point. The motor inertia varies with the attached load, the torque constant has manufacturing tolerance, and the inverter introduces a small computational delay. At high frequencies, current-loop dynamics and mechanical resonances are not represented accurately.

Situation and problem

A nominal PI controller gives satisfactory step-response settling time and negligible steady-state error. However, tests on a heavier load show overshoot and oscillation. Frequency-response comparison indicates that the relative plant error grows substantially beyond the nominal crossover frequency. The design must preserve speed regulation while avoiding excessive torque commands and instability under load variation.

Analysis

The uncertain plant is represented using a nominal model with multiplicative uncertainty:

\[

P=P_0(1+W_m\Delta), \qquad \|\Delta\|_\infty\leq1.

\]

The weight is small at low frequency, where the model is trusted, and increases near the mechanical resonance and inverter-delay region. The nominal loop is examined through \(S\) and \(T\). Although nominal tracking is good, \(T\) peaks near the frequency at which \(W_m\) is already large. Consequently, \(\|W_mT\|_\infty\) exceeds one, so robust stability is not certified.

Approach

The crossover frequency is reduced moderately, a lead element is used to recover phase margin, and a low-pass roll-off is added to limit high-frequency gain. A control-effort weight is included because the torque actuator saturates during rapid speed commands. The resulting mixed-sensitivity problem penalizes low-frequency sensitivity, excessive torque, and high-frequency complementary sensitivity.

The revised design produces a slightly slower nominal response but substantially reduces the sensitivity peak. Worst-case simulations are carried out over the inertia and torque-constant ranges, with delay variation and actuator saturation included. The controller is implemented with anti-windup protection and discrete-time validation at the intended sampling rate.

Outcome

The design meets the required regulation accuracy for the specified load family, avoids the oscillatory response seen with the nominal PI design, and keeps torque demand within the actuator range for the tested commands. The robust certificate does not prove performance outside the uncertainty model; it demonstrates only that the stated assumptions are sufficient for the stated guarantee.

Learning points

The case illustrates that nominal speed response alone is an incomplete design criterion. Uncertainty weighting, sensitivity shaping, actuator constraints, and implementation effects must be considered together. Robust design may intentionally sacrifice nominal speed to obtain predictable behaviour across the operating envelope.

Practical workflow for robust-control design

A defensible workflow starts with requirements stated in measurable terms: bandwidth, settling time, disturbance attenuation, noise level, actuator limits, allowable parameter ranges, and acceptable risk. The next step is to establish the nominal model and validate it against experiments. Uncertainty bounds should be frequency-dependent where evidence supports such a description.

The designer then selects an uncertainty representation and constructs a generalized plant. Initial loop shaping or H2/H-infinity synthesis produces a candidate controller. Analysis includes nominal stability, robust stability, robust performance, sensitivity peaks, control effort, disk margins, and time-domain behaviour. If structured physical variations are important, \(\mu\)-analysis is used rather than relying only on unstructured singular-value bounds.

Controller reduction and implementation checks follow synthesis. High-order controllers may be reduced, but reduction must be accompanied by renewed robustness analysis. Discretization, delays, quantization, sensor filtering, computational scheduling, and saturation can invalidate a continuous-time certificate if ignored. Finally, hardware-in-the-loop and progressively realistic testing should be used before deployment.

Challenges, limitations, and common mistakes

Robust control cannot compensate for a fundamentally incorrect plant model. An uncertainty bound that excludes an important resonance or delay gives a mathematically correct but practically irrelevant result. Conversely, excessively conservative bounds may force low bandwidth and poor performance.

A second difficulty is the conflict among objectives. Tracking, disturbance rejection, noise attenuation, actuator protection, and robustness compete through the relationship between \(S\) and \(T\). No controller can make all channels arbitrarily small over the entire frequency range.

Common mistakes include using parameter ranges without considering correlated variations, treating a nonlinear operating envelope as one local linear model, selecting weights solely to make a synthesis problem feasible, ignoring unstable hidden modes, and interpreting a nominal simulation as a robust guarantee. Another error is to report only the optimized H-infinity value without examining where the worst-case peak occurs or which uncertainty channel causes it.

Robustness analysis also depends on well-posedness and internal stability. Transfer-function cancellation in a nominal model may conceal unstable internal dynamics. Every synthesis and validation exercise should therefore examine internal modes, controller realizability, and numerical conditioning.

Professional, ethical, and governance considerations

Robust-control claims are engineering claims about safety and reliability. The uncertainty set, data sources, assumptions, and excluded conditions should be documented and traceable. In safety-critical systems, independent review, verification and validation, configuration control, and evidence from representative testing are essential. A designer should not describe a controller as “guaranteed” without specifying the model class, norm, uncertainty assumptions, and operating conditions underlying the guarantee.

Automation and optimization software are valuable for synthesis, model fitting, and worst-case search, but they do not replace engineering judgement. Data-driven uncertainty models should preserve physical plausibility and should not hide extrapolation beyond the measured range. When machine-learning models are used inside a control architecture, their uncertainty, update mechanism, failure modes, and supervisory safeguards require explicit governance.

Current developments and emerging directions

Modern robust-control practice increasingly combines frequency-domain theory with data-driven modelling, uncertain linear parameter-varying models, and automated worst-case analysis. Linear fractional transformation modelling provides a common language for connecting physical parameters, dynamic uncertainty, and structured analysis. Robust model predictive control extends uncertainty management to constraints and receding-horizon decision-making, although computational and modelling challenges remain.

Research also considers distributionally robust and risk-sensitive control, in which uncertainty is described probabilistically or through ambiguity sets rather than only by deterministic norm bounds. Networked and cyber-physical systems introduce packet loss, delays, attacks, and changing communication topologies. These developments do not eliminate classical robust-control principles; they broaden the uncertainty channels and make explicit modelling, verification, and resilience assessment even more important.

Best practices

A technically strong robust design normally follows these principles:

• validate the nominal model and document the source of every uncertainty bound;
• use the least conservative uncertainty structure justified by physical knowledge;
• shape sensitivity and complementary sensitivity together rather than optimizing one in isolation;
• include actuator limits, sensor noise, delays, sampling, and unmodelled dynamics early;
• distinguish nominal performance from robust stability and robust performance;
• inspect worst-case frequencies and uncertainty channels, not only scalar norm values;
• reassess guarantees after controller reduction and digital implementation; and
• communicate assumptions and limitations clearly to operators, reviewers, and certification authorities.

Chapter Summary

Robust control designs feedback systems for families of possible plants rather than for a single nominal model. Uncertainty may be parametric, additive, multiplicative, unstructured, or structured. Sensitivity \(S\) and complementary sensitivity \(T\) provide the principal language for understanding disturbance rejection, tracking, noise transmission, and model-error amplification. The small-gain theorem gives norm-based robust-stability conditions such as \(\|W_mT\|_\infty<1\). Structured singular value analysis exploits known uncertainty structure and supports less conservative robust-stability and robust-performance assessment.

H2 control emphasizes integrated energy performance, while H-infinity control limits worst-case peak gain. Mixed-sensitivity synthesis combines tracking, control effort, and high-frequency robustness in one generalized-plant framework. Loop shaping offers a transparent engineering method for selecting bandwidth and roll-off. Effective practice requires validated models, credible uncertainty bounds, implementation-aware analysis, and explicit communication of assumptions. Robustness is not an unlimited property; it is a guarantee relative to a defined uncertainty set and performance specification.

Key Terms

Uncertainty: The discrepancy or lack of knowledge between a mathematical model and the physical plant.

Parametric uncertainty: Variation in known model parameters within specified bounds.

Additive uncertainty: A perturbation added directly to the nominal plant model.

Multiplicative uncertainty: A perturbation that modifies the nominal plant by a relative factor.

Unstructured uncertainty: Dynamic uncertainty represented without detailed internal structure.

Structured uncertainty: Uncertainty whose block arrangement, repetition, or parameter type is known.

Sensitivity function: \(S=(I+PK)^{-1}\), measuring closed-loop sensitivity to several disturbances and perturbations.

Complementary sensitivity: \(T=PK(I+PK)^{-1}\), associated with measurement-noise transmission and closed-loop response.

Sensitivity peak: The maximum value \(\|S\|_\infty\), commonly denoted \(M_s\).

Small-gain theorem: A result guaranteeing stability when the product of interacting system gains is strictly below one.

Robust stability: Stability maintained for every uncertainty in the specified uncertainty set.

Robust performance: Satisfaction of a performance bound for every admissible uncertainty.

Structured singular value: A measure of the smallest structured perturbation capable of causing singularity or instability.

H2 norm: An energy-based norm related to the integrated squared response of a stable system.

H-infinity norm: The peak induced gain, equal to the supremum of the maximum singular value over frequency.

Loop shaping: Design of controller dynamics to produce a desired open-loop frequency response.

Mixed sensitivity: A weighted H-infinity formulation involving \(S\), \(KS\), and \(T\).

Generalized plant: An interconnected model containing plant, uncertainty, exogenous inputs, regulated outputs, and controller interfaces.

D-scaling: A scaling technique used to compute upper bounds for structured singular-value analysis.

Robustness margin: The size of uncertainty that can be tolerated before a specified stability or performance property is lost.

Review Questions

Short Answer Questions

1. Distinguish parametric uncertainty from unstructured dynamic uncertainty.

2. State the relationship between sensitivity and complementary sensitivity functions.

3. What does the condition \(\|W_mT\|_\infty<1\) signify?

4. Why is structured uncertainty often less conservative than unstructured uncertainty?

5. Give one important difference between H2 and H-infinity performance measures.

Long Answer Questions

1. Explain additive and multiplicative uncertainty models, including situations in which each is appropriate.

2. Derive and interpret the role of sensitivity functions in tracking, disturbance rejection, noise attenuation, and robustness.

3. Explain the structured singular value, its robust-stability interpretation, and the purpose of upper and lower bounds.

4. Describe the mixed-sensitivity H-infinity formulation and explain how its three weighting functions are selected.

5. Present a complete workflow for designing, validating, and implementing a robust controller for an uncertain plant.

Analytical / Application Questions

1. A plant has a relative modelling error that increases above 20 rad/s. Explain how the controller should shape \(T\), and identify the likely consequence of placing loop crossover well above that frequency.

2. Two candidate controllers have similar nominal step responses. Controller A has \(M_s=1.4\) and satisfies \(\|W_mT\|_\infty=0.8\); Controller B has \(M_s=2.1\) and \(\|W_mT\|_\infty=1.2\). Which controller provides the stronger stated robust-stability evidence, and why?

3. Design an uncertainty and performance-analysis plan for a motor drive whose inertia varies, whose sensor contains high-frequency noise, and whose actuator saturates. Specify suitable uncertainty representations, performance channels, and validation tests.

22. Multiple Choice Questions with Answers

Question 1

For multiplicative output uncertainty P=P0(I+WmΔ), which condition is a common sufficient robust-stability test?

A. ||WmS||∞ < 1

B. ||WmT||∞ < 1

C. ||KS||∞ > 1

D. ||T||∞ = 0

Correct Answer: B. ||WmT||∞ < 1

Question 2

What does a large sensitivity peak Ms=||S||∞ generally indicate?

A. Improved robustness at every frequency

B. Reduced control effort in all operating conditions

C. A pronounced closed-loop sensitivity and usually smaller stability margins

D. Perfect rejection of measurement noise

Correct Answer: C. A pronounced closed-loop sensitivity and usually smaller stability margins

Question 3

The H-infinity norm of a stable transfer matrix is defined as the:

A. Integral of its impulse response

B. Peak maximum singular value over frequency

C. DC gain only

D. Number of closed-loop poles

Correct Answer: B. Peak maximum singular value over frequency

Question 4

Why can structured singular value analysis be less conservative than an unstructured small-gain test?

A. It ignores all uncertainty

B. It uses the known arrangement and type of uncertainty blocks

C. It assumes the nominal model is exact

D. It applies only to nonlinear systems

Correct Answer: B. It uses the known arrangement and type of uncertainty blocks

Question 5

In mixed-sensitivity design, a weight W1 placed before S is normally made large where the designer wants:

A. High tracking error

B. Low-frequency tracking and disturbance rejection

C. Maximum measurement-noise transmission

D. Unlimited actuator activity

Correct Answer: B. Low-frequency tracking and disturbance rejection

Question 6

Which statement best describes the trade-off between S and T?

A. They are independent at all frequencies

B. S+T=I, so reducing one broadly tends to increase the other somewhere

C. Both can always be made zero with a proper controller

D. T is always equal to S

Correct Answer: B. S+T=I, so reducing one broadly tends to increase the other somewhere

Question 7

A controller has excellent nominal performance but violates ||WmT||∞<1. The most appropriate conclusion is that:

A. Nominal performance proves robust stability

B. The uncertainty model is automatically invalid

C. Robust stability has not been certified for the specified multiplicative uncertainty

D. The plant must be unstable at DC

Correct Answer: C. Robust stability has not been certified for the specified multiplicative uncertainty

Question 8

Which design objective is most directly associated with average energy response under disturbance assumptions?

A. H2 optimization

B. H-infinity optimization

C. Pure phase-margin maximization

D. Structured singular-value maximization

Correct Answer: A. H2 optimization

Question 9

In loop shaping, high-frequency roll-off is primarily used to:

A. Increase amplification of unmodelled dynamics

B. Reduce sensitivity to noise and neglected high-frequency dynamics

C. Eliminate the need for actuator limits

D. Guarantee zero steady-state error without feedback

Correct Answer: B. Reduce sensitivity to noise and neglected high-frequency dynamics

Question 10

After reducing the order of a synthesized robust controller, the designer should:

A. Assume the original robustness certificate remains unchanged

B. Repeat robustness, performance, and implementation analyses

C. Remove all uncertainty weights

D. Evaluate only the nominal step response

Correct Answer: B. Repeat robustness, performance, and implementation analyses