10. Optimal, Robust, and Predictive Control

Introduction and context

Classical feedback control often begins with a nominal mathematical model and a performance objective such as tracking accuracy, disturbance rejection, or stability. That viewpoint is powerful, but advanced mechanical systems rarely operate exactly at their nominal condition. Mass, stiffness, damping, friction, actuator capacity, sensor quality, payload, operating speed, and environmental disturbances may all vary. A controller that performs well for one model can therefore become fragile when the real plant differs from the model used for design.

Optimal, robust, and predictive control provide complementary responses to this difficulty. Optimal control selects actions by minimizing a specified performance index. State estimation, particularly through the Kalman filter, supplies state information when measurements are incomplete or noisy. Robust control explicitly accounts for model uncertainty and seeks acceptable performance across a family of plants. Predictive control repeatedly solves an optimization problem using a model of future system behavior while respecting constraints.

These methods are not unrelated alternatives. An industrial controller may use a Kalman filter to estimate states, an LQR law for nominal feedback, robust loop-shaping to protect stability margins, and model predictive control to handle actuator and safety constraints. The central engineering task is to decide how much model detail, computation, uncertainty description, and performance ambition are justified by the application.

Learning objectives

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

1. formulate linear-quadratic regulator and linear-quadratic-Gaussian control problems;

2. explain the role of the Riccati equation and the separation principle;

3. construct and interpret Kalman-filter models;

4. distinguish parametric, unstructured, stochastic, and bounded uncertainty;

5. explain robust stability, robust performance, sensitivity, complementary sensitivity, and H-infinity design;

6. use loop-shaping concepts to balance bandwidth, disturbance rejection, noise attenuation, and robustness;

7. formulate a finite-horizon model predictive control problem with constraints;

8. interpret the performance–robustness trade-off in mechanical systems;

9. identify implementation risks, validation requirements, and professional responsibilities in advanced control design.

From nominal feedback to decision-making under uncertainty

Consider a continuous-time linear model

\[

\dot{x}=Ax+Bu+Ew, \qquad y=Cx+v,

\]

where \(x\) is the state, \(u\) the control input, \(w\) a process disturbance, and \(v\) measurement noise. In a nominal design, matrices \(A\), \(B\), and \(C\) are treated as exact. In practice, they represent estimates. A flexible structure may have uncertain modal frequencies; a robot may carry different payloads; a servo drive may experience saturation and friction; and a vehicle suspension may operate over a wide range of speeds.

Three questions distinguish the advanced control approaches in this chapter:

• Optimal control: among admissible inputs, which minimizes a defined cost?
• Robust control: which controller maintains stability and acceptable performance despite uncertainty?
• Predictive control: which present action is best when future plant behavior and constraints are considered over a finite horizon?

The word optimal must always be qualified. A controller is optimal only relative to its model, cost function, constraints, and assumed information. Robustness is also not a universal property: a design is robust with respect to a stated uncertainty set and specified performance requirements.

LQR: optimal regulation for a nominal model

For the system

\[

\dot{x}=Ax+Bu,

\]

linear-quadratic regulation chooses a state-feedback law to minimize

\[

J=\int_0^\infty (x^TQx+u^TRu)\,dt,

\]

where \(Q\succeq0\) penalizes state deviations and \(R\succ0\) penalizes control effort. Cross-terms may be included, but the basic form already illustrates the design logic. Large entries in \(Q\) demand rapid reduction of corresponding states; large entries in \(R\) discourage aggressive actuator use.

Under standard stabilizability and detectability conditions, the solution has the form

\[

u=-Kx, \qquad K=R^{-1}B^TP,

\]

where \(P\) is the positive semidefinite solution of the continuous algebraic Riccati equation (CARE),

\[

A^TP+PA-PBR^{-1}B^TP+Q=0.

\]

The Riccati equation is not merely a computational device. It expresses the balance between the natural dynamics, state penalties, and control cost. The resulting closed-loop matrix is \(A-BK\). Unlike pole placement, LQR does not directly prescribe individual poles; it produces a coordinated placement governed by the cost function and system structure.

Weight selection and scaling

Weight selection is a modeling decision rather than a trial-and-error ritual. A useful starting point is Bryson’s rule: if \(x_{i,\max}\) and \(u_{j,\max}\) are acceptable maximum values, choose approximate diagonal weights

\[

Q_{ii}=1/x_{i,\max}^2, \qquad R_{jj}=1/u_{j,\max}^2.

\]

The variables should be scaled so that numerical magnitudes do not obscure physical priorities. Increasing state weights usually increases bandwidth and control activity, but the result also depends on plant zeros, actuator dynamics, sampling, and unmodeled modes. Weight changes must therefore be assessed using time responses, frequency responses, actuator demands, and robustness margins.

For tracking rather than regulation, reference precompensation, integral augmentation, or an internal-model structure may be required. Integral action is especially important when constant disturbances or model bias would otherwise produce steady-state error.

LQG and Kalman state estimation

Full-state feedback is often impractical because velocities, flexible-mode coordinates, forces, or temperatures may not be directly measured. The Kalman filter estimates the state using a model and noisy measurements. In discrete time,

\[

x_{k+1}=Ax_k+Bu_k+w_k,\qquad y_k=Cx_k+v_k,

\]

with process-noise covariance \(W\) and measurement-noise covariance \(V\). The filter alternates between prediction and correction:

\[

\hat{x}_{k|k-1}=A\hat{x}_{k-1|k-1}+Bu_{k-1},

\]

\[

\hat{x}_{k|k}=\hat{x}_{k|k-1}+L_k(y_k-C\hat{x}_{k|k-1}).

\]

The gain \(L_k\) is calculated from the covariance recursion. Large assumed process uncertainty makes the estimator rely more strongly on measurements; large measurement-noise covariance makes it rely more on the model. Incorrect covariance assumptions can cause sluggish estimation, excessive noise transmission, or inconsistency between predicted and observed behavior.

LQG combines an LQR controller with a Kalman filter:

\[

u_k=-K\hat{x}_k.

\]

For the ideal linear-Gaussian problem, the separation principle allows controller and estimator gains to be designed independently. This does not imply unconditional robustness. LQG can exhibit poor stability margins when the model is uncertain, even though the nominal closed loop is optimal in a statistical sense. Practical LQG designs therefore require robustness checks, sensor filtering analysis, and validation against non-Gaussian disturbances, saturation, delay, and unmodeled dynamics.

Modeling uncertainty: what is unknown and how it enters

Uncertainty should be described according to its source and structure. Parametric uncertainty concerns unknown physical parameters, such as mass, damping, or stiffness. It can be represented by intervals, probability distributions, or a finite set of operating-point models. Unstructured uncertainty represents neglected dynamics or high-frequency effects without assuming a detailed physical form. Structured uncertainty preserves known relationships among uncertain parameters or repeated blocks.

A common multiplicative output uncertainty model is

\[

G(s)=G_0(s)[1+W_m(s)\Delta(s)], \qquad \|\Delta\|_\infty\le1,

\]

where \(G_0\) is the nominal plant and \(W_m\) describes the frequency-dependent uncertainty magnitude. Uncertainty is often small at low frequency, where the plant has been identified carefully, and larger at high frequency, where flexible modes, delays, and actuator dynamics are neglected.

The uncertainty weight must not be selected merely to make analysis convenient. It should be supported by experiments, identification residuals, physical bounds, or a deliberately conservative engineering argument. An uncertainty description that is too small gives false confidence; one that is excessively large may make a useful controller appear impossible.

Sensitivity, loop-shaping, and robust stability

For a single-input, single-output loop with loop transfer function \(L(s)=G(s)K(s)\), the sensitivity and complementary sensitivity functions are

\[

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

\]

Sensitivity governs the response of the closed loop to reference and disturbance effects at several points in the loop. Complementary sensitivity describes how measurement noise and high-frequency dynamics can pass to the output. The identity \(S+T=1\) shows why performance objectives conflict: reducing sensitivity over one frequency range generally increases it elsewhere.

A typical loop-shaping objective is high loop gain at low frequencies for tracking and disturbance rejection, a crossover region with adequate phase and gain margin, and low gain at high frequencies to attenuate noise and avoid exciting unmodeled modes. Lead compensation can recover phase near crossover; lag compensation can increase low-frequency gain with limited effect on crossover; notch filters can suppress known resonances, although they may introduce sensitivity to frequency shifts.

For multiplicative output uncertainty, a sufficient robust-stability condition is commonly expressed as

\[

\|W_mT\|_\infty<1.

\]

This means that the closed loop must not amplify the uncertainty weight to a level that permits an unstable perturbed plant. The condition provides a frequency-domain interpretation: where the model is uncertain, complementary sensitivity must be correspondingly small.

H-infinity control and the performance–robustness compromise

H-infinity design treats the worst-case gain from exogenous inputs to regulated outputs as the quantity to be minimized. A generalized plant may contain disturbances, references, measurement noise, control signals, and weighted performance outputs. The controller is selected so that

\[

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

\]

where \(w\) represents exogenous inputs and \(z\) weighted regulated outputs. The value \(\gamma\) bounds the largest induced energy gain over frequency.

Weights translate engineering requirements into mathematical form. A low-frequency performance weight can demand small tracking error; a high-frequency control weight can limit actuator activity; and an uncertainty-related weight can constrain sensitivity near neglected modes. H-infinity synthesis often gives a controller that is more conservative than LQR, but it explicitly addresses worst-case behavior rather than average or nominal behavior.

The design parameter \(\gamma\) reflects a feasibility frontier. Requiring a very small \(\gamma\) may produce high-order controllers, excessive control action, poor numerical conditioning, or no feasible solution. Accepting a larger \(\gamma\) may improve implementability while reducing disturbance attenuation. Controller reduction, balanced truncation, and structured implementation are frequently needed before deployment.

Robust performance includes both robust stability and satisfactory response. A controller that never becomes unstable but has unacceptable tracking error is not a successful robust design. Conversely, a controller with excellent nominal response but narrow stability margins may be unsuitable for production equipment.

Model predictive control: optimization over a moving horizon

Model predictive control (MPC) uses a model to predict future outputs and selects a sequence of future control actions by solving an optimization problem at each sampling instant. Only the first action is implemented; the horizon then moves forward and the problem is solved again using updated measurements or estimates.

A linear discrete-time MPC problem can be written as

\[

\min_{u_0,\ldots,u_{N-1}} \sum_{i=0}^{N-1}\left[(x_i-r_i)^TQ(x_i-r_i)+u_i^TRu_i\right]

+(x_N-r_N)^TP_f(x_N-r_N),

\]

subject to

\[

x_{i+1}=Ax_i+Bu_i,

\]

and constraints such as

\[

x_i\in\mathcal X,\qquad u_i\in\mathcal U,\qquad \Delta u_i\in\mathcal D.

\]

The terminal weight \(P_f\), terminal set, and horizon are selected to support stability and acceptable long-term behavior. For linear-quadratic unconstrained MPC, the solution approaches an LQR law when the horizon and terminal design are suitable. Thus MPC can be viewed as an extension of optimal control that incorporates finite-horizon preview and constraints.

Constraints are often the principal reason to use MPC. They may represent motor current, valve position, acceleration, temperature, travel limits, contact force, or structural stress. Hard constraints must never be violated, while soft constraints are relaxed using penalty variables when temporary violation is safer than optimization failure. Poor scaling, infeasible references, inaccurate state estimates, and insufficient computation time are common implementation problems.

MPC may be linear, nonlinear, robust, stochastic, or economic. Linear MPC is attractive for real-time implementation and can be formulated as a quadratic program. Nonlinear MPC can represent large motion, nonlinear friction, contact, or thermodynamic behavior, but it imposes heavier computational and verification demands. Explicit MPC precomputes a piecewise-affine control law for certain constrained linear problems, trading memory for online speed.

A practical design workflow

A defensible advanced-control project normally follows a staged process:

1. Define requirements. Specify tracking error, settling time, allowable overshoot, disturbance levels, actuator limits, safety constraints, sampling rate, and acceptable computational delay.

2. Develop and validate models. Combine first-principles modeling with system identification. Test the model over operating conditions rather than only at one nominal point.

3. Quantify uncertainty. Separate parameter variation, measurement noise, neglected dynamics, delays, and nonlinear effects. Select uncertainty weights or scenarios supported by evidence.

4. Choose the architecture. LQR/LQG may suit unconstrained regulation; robust loop-shaping may be preferred where uncertainty dominates; MPC is attractive when constraints and preview are central; hybrid architectures are common.

5. Scale and tune objectives. Select physically meaningful weights and verify the effect of each weight on performance and effort.

6. Analyze nominal and robust behavior. Examine poles, singular values, sensitivity functions, gain and phase margins, worst-case perturbations, actuator demand, and noise response.

7. Test nonidealities. Include saturation, rate limits, quantization, delays, sensor dropout, initialization errors, friction, backlash, and sampling effects.

8. Validate progressively. Move from simulation to software-in-the-loop, hardware-in-the-loop, component testing, and controlled system trials.

9. Monitor after commissioning. Record operating data, detect drift, and define conditions requiring retuning or safe fallback.

Illustrative case study: robust predictive control of a precision positioning stage

Background

Consider a two-axis electromechanical positioning stage used to place components on a manufacturing line. Each axis is driven by a permanent-magnet motor and ball screw. The nominal model includes position, velocity, motor current, and a dominant structural resonance. Payload changes alter the effective mass, while resonance frequency varies with fixture configuration.

Situation and problem

A nominal LQR controller provides rapid positioning for the lightest payload, but heavier payloads produce overshoot and longer settling time. Increasing the state weights improves the nominal response but drives current close to its limit and excites the structural resonance. An LQG implementation reduces encoder-noise effects, yet tests reveal poor gain margin when the resonance shifts. The stage must satisfy position limits, motor-current limits, and a maximum acceleration requirement.

Analysis

Identification experiments produce a family of models indexed by payload and fixture stiffness. The low-frequency dynamics are well known, but uncertainty grows near the resonance and above the current-loop bandwidth. The engineering objectives are therefore: strong low-frequency disturbance rejection, limited complementary sensitivity near the uncertain resonance, bounded current and acceleration, and acceptable computational delay.

Approach and solution

First, the states and control signals are scaled using allowable position, velocity, current, and acceleration values. An LQR design establishes a nominal performance benchmark. A Kalman filter estimates velocity and the flexible coordinate from encoder position and motor-current measurements.

Next, multiplicative uncertainty is fitted around the family of identified models. A robust loop-shaping analysis shows that the nominal LQG controller has excessive complementary sensitivity near the uncertain resonance. A revised controller uses a lower crossover frequency, a resonance notch with sufficient bandwidth, and a high-frequency roll-off. The resulting nominal response is slower but has improved worst-case stability margins.

Finally, a constrained linear MPC layer operates at the motion-planning level. It predicts position and velocity over a finite horizon, enforces current and acceleration constraints, and supplies feasible reference trajectories to the robust servo loop. Soft position-tracking constraints prevent optimization failure during an infeasible command, while hard current and travel limits remain enforced.

Outcome

The combined architecture sacrifices some nominal settling speed but avoids current saturation and remains stable across the modeled payload range. Simulation and hardware-in-the-loop tests include resonance shifts, sensor noise, timing jitter, and command changes. Before production release, the design requires controlled tests at payload extremes and a fallback mode that disables the predictive layer if its solver exceeds the timing budget.

Learning points

The case demonstrates that no single method solves every problem. LQR supplies a useful nominal benchmark, Kalman filtering improves state information, robust design protects against model variation, and MPC handles constraints and command feasibility. The final architecture is chosen from system requirements rather than from the prestige of a particular algorithm.

Challenges, risks, and professional responsibilities

Advanced controllers can fail for reasons unrelated to the mathematical optimization. A sensor bias may be interpreted as a physical displacement; actuator saturation may invalidate a linear design; computational delay may destroy phase margin; and an estimator may diverge after an unexpected mode is excited. Discrete implementation must account for sampling, zero-order hold, numerical precision, task scheduling, and communication delays.

Common mistakes include using arbitrary, unscaled weights; claiming robustness without specifying an uncertainty set; validating only the nominal model; ignoring actuator rate limits; assuming that a Kalman filter automatically removes all noise; and treating a feasible simulation as proof of safe operation. A controller should also be checked for recoverability after power interruptions, sensor faults, and initialization errors.

Control software is part of a safety-relevant system when it governs machinery, vehicles, medical equipment, or industrial processes. Requirements must be traceable, assumptions documented, changes controlled, and test evidence retained. A robust-control certificate is not a substitute for physical guarding, independent safety functions, emergency stops, or compliance with applicable machinery and functional-safety requirements. Data used for adaptive or learning-based tuning must be governed for integrity, traceability, and cybersecurity.

Digital implementation and emerging directions

Modern control design relies heavily on numerical optimization, automatic differentiation, real-time operating systems, high-fidelity simulation, and hardware-in-the-loop testing. Digital twins can support uncertainty studies when they are calibrated against measured data, but they should not be treated as unquestionable replicas of the physical plant. Online estimation and fault detection can identify parameter drift, while cloud connectivity may support fleet-level analysis; safety-critical control should nevertheless preserve deterministic local operation.

Current developments include distributionally robust MPC, chance-constrained control, tube MPC, data-enabled modeling, learning-based residual dynamics, and reinforcement-learning policies constrained by a verified backup controller. These methods may improve performance under changing conditions, but they introduce new questions about data coverage, explainability, verification, and out-of-distribution behavior. For postgraduate engineers, the important trend is not replacement of physical modeling by data alone, but integration of physics, uncertainty quantification, optimization, and dependable computation.

Best-practice principles

A reliable advanced-control design should:

• state performance, uncertainty, and safety requirements quantitatively;
• use physically scaled variables and document all weighting choices;
• distinguish nominal optimality from robust acceptability;
• validate uncertainty descriptions with experiments or defensible bounds;
• analyze sensitivity and actuator demand, not only output plots;
• include constraints, delays, saturation, and estimator failure modes;
• retain a simple, testable fallback controller;
• verify real-time execution and numerical conditioning;
• test at operating extremes and during transitions between regimes;
• monitor performance after deployment and maintain configuration control.

Chapter Summary

LQR minimizes a quadratic performance index for a nominal state-space model and produces a Riccati-equation-based state-feedback law. LQG combines LQR with Kalman estimation, but nominal stochastic optimality does not guarantee robust stability. Uncertainty must be modeled according to its physical source and frequency structure. Sensitivity and complementary sensitivity provide a language for understanding tracking, disturbance rejection, noise transmission, and robustness. H-infinity design addresses worst-case induced gain and uses weighting functions to express performance and uncertainty requirements. Loop-shaping balances low-frequency performance against high-frequency robustness.

MPC repeatedly solves a finite-horizon optimization problem and is especially valuable when input, state, rate, and safety constraints are important. Its design requires suitable models, terminal or stability considerations, reliable state estimates, and guaranteed computational timing. In practice, LQR/LQG, robust control, and MPC are often combined. The correct choice depends on uncertainty, constraints, computation, safety, and the evidence available for validation. Performance must always be interpreted together with robustness, implementability, and professional responsibility.

Key Terms

LQR: State-feedback control that minimizes a quadratic cost for a linear model.

CARE: Continuous algebraic Riccati equation used to compute the infinite-horizon LQR solution.

LQG: Combination of LQR state feedback and Kalman state estimation.

Kalman filter: Recursive minimum-variance state estimator for a specified stochastic linear model.

Separation principle: Property allowing nominal controller and estimator gains to be designed independently under stated assumptions.

Parametric uncertainty: Variation in identifiable physical parameters such as mass or stiffness.

Unstructured uncertainty: Uncertainty representing neglected dynamics without detailed internal structure.

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

Robust performance: Satisfaction of performance requirements for every allowed uncertain plant.

Sensitivity function: \(S=(1+L)^{-1}\), describing the closed-loop sensitivity to several disturbances.

Complementary sensitivity: \(T=L(1+L)^{-1}\), commonly associated with noise transmission and high-frequency behavior.

Loop-shaping: Frequency-domain design of loop gain to achieve desired performance and margins.

H-infinity norm: Peak singular-value or gain measure over frequency.

MPC: Control based on repeated finite-horizon prediction and optimization.

Prediction horizon: Number of future sampling intervals included in an MPC optimization.

Soft constraint: Constraint permitted to relax through a penalized slack variable.

Terminal cost: End-of-horizon penalty used to represent future behavior and support MPC stability.

Tube MPC: Robust MPC method that maintains predicted trajectories inside uncertainty-related tubes.

Review Questions

Short Answer Questions

1. What physical meaning do the matrices \(Q\) and \(R\) have in an LQR problem?

2. Why is a Kalman filter required in an LQG controller?

3. Distinguish robust stability from robust performance.

4. What are the sensitivity and complementary sensitivity functions?

5. Why are constraints a primary motivation for using MPC?

Long Answer Questions

1. Derive the LQR state-feedback structure and explain the role of the continuous algebraic Riccati equation.

2. Explain the prediction and correction steps of a discrete-time Kalman filter and discuss the effect of process-noise and measurement-noise covariances.

3. Discuss how uncertainty weights are constructed and used in robust stability analysis.

4. Compare LQR, H-infinity loop-shaping, and MPC with respect to objectives, uncertainty, constraints, and computational requirements.

5. Explain a complete workflow for designing and validating an advanced controller for a mechanical system.

Analytical / Application Questions

1. A servo system has excellent nominal LQG performance but a flexible mode whose frequency varies by 20 percent. Explain which frequency-domain quantities should be examined and how the controller might be modified.

2. Formulate an MPC problem for a motorized positioning axis with position, velocity, current, and current-rate limits. Identify which constraints should be hard and which might reasonably be soft.

3. A plant has multiplicative uncertainty described by \(W_m(s)\). Explain how the condition \(\|W_mT\|_\infty<1\) influences crossover frequency and high-frequency controller roll-off.

22. Multiple Choice Questions with Answers

Question 1

In an LQR design, increasing the control-weight matrix R generally tends to:

A. penalize control effort more strongly and produce less aggressive input

B. eliminate the need for state measurements

C. increase model uncertainty automatically

D. force all closed-loop poles to be identical

Correct Answer: A. penalize control effort more strongly and produce less aggressive input

Question 2

In a Kalman filter, increasing the assumed measurement-noise covariance V generally causes the estimator to:

A. trust measurements more strongly

B. rely more heavily on the model prediction

C. remove process disturbances from the plant

D. increase actuator bandwidth

Correct Answer: B. rely more heavily on the model prediction

Question 3

Which statement best describes the limitation of nominal LQG design?

A. It cannot use a state estimator

B. It guarantees robust performance for every uncertainty set

C. It may have poor stability margins despite nominal stochastic optimality

D. It is applicable only to nonlinear plants

Correct Answer: C. It may have poor stability margins despite nominal stochastic optimality

Question 4

For a feedback loop, the complementary sensitivity function is:

A. S=1+L

B. S=(1+L)^−1

C. T=L(1+L)^−1

D. T=1/L

Correct Answer: C. T=L(1+L)^−1

Question 5

For multiplicative output uncertainty represented by WmΔ, a common sufficient robust-stability condition is:

A. ||WmS||∞>1

B. ||WmT||∞<1

C. ||T||∞=0 at all frequencies

D. ||S||∞<0

Correct Answer: B. ||WmT||∞<1

Question 6

Which feature most directly distinguishes MPC from unconstrained infinite-horizon LQR?

A. MPC uses no plant model

B. MPC repeatedly optimizes predicted future actions subject to constraints

C. MPC cannot penalize control effort

D. MPC requires every state to be measured directly

Correct Answer: B. MPC repeatedly optimizes predicted future actions subject to constraints

Question 7

A robust controller reduces gain near a poorly modeled structural resonance primarily to:

A. increase complementary sensitivity at that resonance

B. reduce the effect of uncertainty and avoid exciting neglected dynamics

C. remove all low-frequency tracking capability

D. make the actuator unlimited

Correct Answer: B. reduce the effect of uncertainty and avoid exciting neglected dynamics

Question 8

In MPC, a soft constraint is implemented by:

A. removing the constraint from the model without replacement

B. adding a penalized slack variable that permits controlled violation

C. forcing the prediction horizon to zero

D. replacing the objective with a pole-placement equation

Correct Answer: B. adding a penalized slack variable that permits controlled violation

Question 9

Why are physical scaling and Bryson-type weight selection useful in optimal control?

A. They make all plants exactly linear

B. They relate numerical weights to acceptable physical state and input magnitudes

C. They guarantee robust stability

D. They eliminate measurement noise

Correct Answer: B. They relate numerical weights to acceptable physical state and input magnitudes

Question 10

In the positioning-stage case, combining robust servo control with MPC is justified because:

A. MPC alone always guarantees stability under unmodeled dynamics

B. robust control addresses model variation while MPC handles operating constraints and feasible commands

C. the Kalman filter replaces all actuator constraints

D. LQR is only valid for systems without actuators

Correct Answer: B. robust control addresses model variation while MPC handles operating constraints and feasible commands