9. Optimal Control and Decision Frameworks

Introduction and context

Classical control design often begins with a stability requirement and proceeds toward acceptable transient response, disturbance rejection, and robustness. Advanced engineering problems, however, rarely have a single performance requirement. An electric-drive controller may need to track speed accurately, limit current, reduce copper losses, avoid actuator saturation, and preserve equipment life. A power-system controller may have to balance frequency quality, reserve utilization, control effort, and the consequences of rare but severe disturbances. These requirements cannot be handled adequately by asking only whether the closed-loop system is stable.

Optimal control provides a mathematical framework for selecting, among admissible control actions, the one that minimizes a specified performance index. The word “optimal” therefore has meaning only relative to an objective, a model, constraints, and a time horizon. Decision frameworks extend this idea by recognizing that engineering choices involve multiple objectives, uncertainty, risk, lifecycle cost, and sometimes qualitative judgments. The central professional task is not merely to compute a gain matrix; it is to formulate a defensible decision problem and understand the implications of its solution.

This chapter develops the principal ideas behind dynamic programming, the calculus of variations, linear-quadratic regulation (LQR), and linear-quadratic-Gaussian (LQG) control. It also considers energy, control effort, risk, lifecycle cost, multi-objective trade-offs, implementation constraints, and emerging computational methods.

Learning objectives

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

1. formulate performance indices, boundary conditions, and constraints for optimal-control problems;

2. distinguish finite-horizon, infinite-horizon, open-loop, closed-loop, deterministic, and stochastic formulations;

3. explain the relationship between the calculus of variations, Bellman’s principle, the Hamilton–Jacobi–Bellman equation, and the Riccati equation;

4. derive and interpret the continuous-time LQR solution;

5. explain how state estimation leads to the LQG controller and the separation principle;

6. evaluate control effort, energy, risk, reliability, and lifecycle cost in control decisions;

7. identify limitations caused by model uncertainty, constraints, nonlinearity, and poor objective formulation; and

8. select and justify an appropriate optimal-control or decision framework for an engineering application.

9.1 From performance requirements to an optimization problem

Consider a system described by

\[

\dot{x}(t)=f(x(t),u(t),t), \qquad y(t)=h(x(t),u(t),t),

\]

where \(x\) is the state, \(u\) the control input, and \(y\) the measured output. An optimal-control problem specifies a set of admissible controls \(\mathcal U\), system constraints, boundary conditions, and a cost functional. A common finite-horizon objective is

\[

J=\Phi(x(T),T)+\int_{t_0}^{T} L(x(t),u(t),t)\,dt.

\]

The running cost \(L\) measures performance accumulated during operation, while the terminal cost \(\Phi\) penalizes the final state. For a motor, \(L\) may include speed error and copper loss; for a battery system, it may include state-of-charge deviation, current magnitude, and degradation cost.

A quadratic tracking objective can be written as

\[

J=\frac{1}{2}(x(T)-x_r)^T F(x(T)-x_r)

+\frac{1}{2}\int_{0}^{T}\left[(x-x_r)^TQ(x-x_r)+u^TRu\right]dt.

\]

Here, \(Q\succeq0\) weights state error, \(R\succ0\) weights control effort, and \(F\succeq0\) weights terminal error. The weighting matrices are not arbitrary mathematical decorations. They encode engineering priorities and units. Scaling, normalization, and sensitivity studies are therefore essential before interpreting a computed controller.

Constraints may be imposed on states, inputs, rates, outputs, and safety margins:

\[

u_{\min}\leq u(t)\leq u_{\max},\qquad x(t)\in\mathcal X,

\]

or through inequalities such as temperature, current, voltage, and actuator-rate limits. An unconstrained LQR solution may demand an input that cannot be physically delivered. Consequently, the unconstrained optimum is not automatically the implementable optimum.

Important terminology includes the following distinctions. An open-loop optimal control is computed from the initial condition and applied without using subsequent measurements. A feedback or closed-loop policy maps the current state, estimate, or measurement history into a control action. A finite-horizon problem has a specified terminal time; an infinite-horizon problem evaluates performance indefinitely, often with a steady-state policy. A deterministic problem assumes known dynamics and disturbances, whereas a stochastic problem explicitly models random variables or processes.

9.2 Variational reasoning and necessary conditions

The calculus of variations addresses the problem of selecting a function, rather than a finite-dimensional vector, to minimize a functional. For a simple functional

\[

J[x]=\int_{t_0}^{T} L(x,\dot{x},t)dt,

\]

a necessary condition for an interior optimum is the Euler–Lagrange equation

\[

\frac{\partial L}{\partial x}-\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right)=0.

\]

In control, the state equation is a constraint. Introduce a costate \(\lambda(t)\) and define the Hamiltonian

\[

H(x,u,\lambda,t)=L(x,u,t)+\lambda^Tf(x,u,t).

\]

For a sufficiently regular optimum, Pontryagin’s minimum principle gives

\[

\dot{x}=\frac{\partial H}{\partial\lambda}=f(x,u,t),

\]

\[

\dot{\lambda}=-\frac{\partial H}{\partial x},

\]

and the minimizing control satisfies

\[

u^(t)=\arg\min_{u\in\mathcal U}H(x^,u,\lambda^*,t).

\]

These conditions are generally necessary, not sufficient. They produce a two-point boundary-value problem: the initial state is usually known, while the terminal costate is determined by the terminal condition. Numerical solution can be difficult, especially for nonlinear or unstable systems.

The costate is often interpreted as a marginal value. Its components indicate how a small change in the corresponding state affects the minimum future cost. This interpretation is useful in engineering decisions: a high costate associated with battery temperature, for example, signals that thermal state has substantial economic or operational significance at that time.

9.3 Dynamic programming and the value function

Dynamic programming uses Bellman’s principle of optimality: an optimal policy has the property that, regardless of the state reached at an intermediate time, the remaining policy must be optimal for the remaining problem. Define the value function

\[

V(x,t)=\min_{u(\cdot)}\left[\Phi(x(T),T)+\int_t^T L(x(\tau),u(\tau),\tau)d\tau\right].

\]

For continuous deterministic systems, Bellman’s principle leads to the Hamilton–Jacobi–Bellman (HJB) equation

\[

-\frac{\partial V}{\partial t}=\min_u\left[L(x,u,t)+\nabla_xV^T f(x,u,t)\right],

\]

with terminal condition \(V(x,T)=\Phi(x,T)\). The optimal control is obtained by minimizing the expression inside the brackets.

Dynamic programming produces a feedback policy and is conceptually powerful because it handles the future consequences of present decisions. Its principal difficulty is computational: the value function depends on every state variable. If each state dimension is discretized into \(N\) points, a grid may require approximately \(N^n\) points for an \(n\)-state system. This “curse of dimensionality” explains why exact dynamic programming is mainly used for low-dimensional systems, special structures, approximate methods, or offline policy generation.

The calculus-of-variations and dynamic-programming viewpoints are complementary. Pontryagin’s principle describes local necessary conditions along an optimal trajectory. Dynamic programming describes a global cost-to-go function and directly yields feedback. When the value function is differentiable, the relation

\[

\lambda(t)=\nabla_xV(x(t),t)

\]

connects the costate to the value gradient.

9.4 Linear-quadratic regulation

For the linear system

\[

\dot{x}=Ax+Bu,

\]

consider the infinite-horizon cost

\[

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

\]

where \(Q\succeq0\) and \(R\succ0\). The LQR problem seeks a stabilizing state-feedback law that minimizes this cost. Assume a quadratic value function

\[

V(x)=x^TPx,

\]

where \(P=P^T\succeq0\). Substitution into the HJB equation gives

\[

0=\min_u\left[x^TQx+u^TRu+2x^TP(Ax+Bu)\right].

\]

Differentiating with respect to \(u\) gives

\[

2Ru+2B^TPx=0,

\]

and therefore

\[

u^*=-R^{-1}B^TPx=-Kx.

\]

The matrix \(P\) satisfies the continuous-time algebraic Riccati equation (CARE):

\[

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

\]

The feedback gain is

\[

K=R^{-1}B^TP.

\]

For standard existence results, the pair \((A,B)

\) must be stabilizable and the pair \((Q^{1/2},A)\) detectable. Under these conditions, the stabilizing solution makes \(A-BK\) asymptotically stable. The value \(x^TPx\) is not merely a computational device; it represents the minimum future cost associated with state \(x\).

The weighting matrices determine behavior. Increasing \(Q\) relative to \(R\) generally produces more aggressive state regulation, while increasing \(R\) penalizes actuator activity and usually produces a less forceful response. This statement is qualitative rather than universal because changes in one weight can interact with system dynamics and other weights. Cross-weight terms may also be used, although they require care to preserve a well-posed problem.

For a finite horizon, the value function is \(V(x,t)=x^TP(t)x\), and \(P(t)\) satisfies the differential Riccati equation

\[

-\dot P=A^TP+PA-PBR^{-1}B^TP+Q,

\]

with terminal condition \(P(T)=F\). The resulting gain is time-varying: \(K(t)=R^{-1}B^TP(t)\). A finite-horizon controller can be preferable when terminal performance matters, such as maneuver completion, energy scheduling, or constrained mission planning.

Interpreting control effort and energy

The term \(u^TRu\) is commonly called control effort. Its physical interpretation depends on the actuator. For a voltage-driven motor, it may correlate with electrical loss, but it is not necessarily identical to energy consumption. True energy may require an integral such as

\[

E=\int_0^T p(t)dt,

\]

where \(p(t)\) is electrical or mechanical power. If regeneration, efficiency variation, or asymmetric actuator limits are important, a simple quadratic effort term may be inadequate. A sound objective distinguishes mathematical regularization from a physically validated energy model.

9.5 LQG: optimal control with incomplete state information

LQR assumes that the complete state is available. In practice, measurements are noisy and some states are not directly measured. The linear stochastic model is

\[

dx=Ax\,dt+Bu\,dt+G\,dw,

\]

\[

dy=Cx\,dt+D\,dv,

\]

where \(w\) and \(v\) are zero-mean white-noise processes with covariance matrices related to \(W\) and \(V\). The Kalman filter estimates the state using the model and measurements:

\[

\dot{\hat{x}}=A\hat{x}+Bu+L_f(y-C\hat{x}),

\]

where \(L_f\) is the estimator gain. Its covariance is obtained from a Riccati equation, and the steady-state gain is

\[

L_f=\Sigma C^TV^{-1}

\]

for the basic continuous-time measurement model, where \(\Sigma\) solves the estimator algebraic Riccati equation.

The LQG controller applies the LQR law to the estimated state:

\[

u=-K\hat{x}.

\]

Under the assumptions of linear dynamics, Gaussian disturbances, quadratic cost, and appropriate stabilizability and detectability conditions, the separation principle states that the controller gain and estimator gain can be designed independently. This does not mean that estimation and control are unrelated in performance; it means that their optimal design decomposes mathematically under the stated assumptions.

LQG has important limitations. Its nominal optimality may be fragile under unmodeled dynamics, non-Gaussian disturbances, actuator saturation, and hard safety constraints. A controller designed from a stochastic average-cost criterion may perform poorly when rare events dominate risk. Robust control, constrained predictive control, risk-sensitive control, or fault-tolerant architectures may then be more suitable.

9.6 Decision frameworks beyond a single quadratic cost

Engineering decisions frequently involve objectives that cannot be reduced naturally to one number. Typical objectives include tracking quality, energy consumption, peak actuator demand, reliability, thermal stress, maintenance cost, emissions, and response to low-probability high-consequence events. A multi-objective formulation may be written as

\[

J(\theta)=\left[J_1(\theta),J_2(\theta),\ldots,J_m(\theta)\right],

\]

where \(\theta\) represents controller parameters or a policy. One common scalarization is

\[

J_w=\sum_{i=1}^{m}w_i\tilde J_i,

\]

where normalized objectives \(\tilde J_i\) and nonnegative weights \(w_i\) are selected by stakeholders. Weight selection should be documented because it can change the preferred design.

A Pareto-optimal solution is one for which no objective can be improved without worsening at least one other objective. The Pareto frontier is often more informative than a single selected point. For example, a modest increase in energy use may greatly reduce settling time, whereas further increases may yield little benefit. Such a frontier reveals diminishing returns and supports transparent decisions.

Constraints and objectives must be separated carefully. A legal voltage limit or protection threshold is normally a hard constraint, not merely a heavily weighted penalty. Soft constraints may be represented by slack variables with explicit violation costs, but safety-critical violations should not be hidden inside an arbitrary objective function.

Risk-sensitive evaluation can include expected cost, variance, probability of constraint violation, conditional value-at-risk, or worst-case loss. Expected cost alone may favor a controller with excellent average behavior but unacceptable tail behavior. Lifecycle cost expands the horizon beyond immediate control performance:

\[

C_{\text{life}}=C_0+\sum_{k=1}^{N}\frac{C_k^{\text{energy}}+C_k^{\text{maintenance}}+C_k^{\text{failure}}}{(1+r)^k},

\]

where \(r\) is a discount rate. The model should include uncertainty in degradation, replacement, downtime, and demand rather than implying unjustified precision.

9.7 Applied scenario: energy-aware control of an industrial drive

The following is an illustrative case. A variable-speed pump in a process plant must maintain flow despite changing demand. The drive has a measured speed, estimated motor flux, and a limited inverter current. Excessive current causes heating and accelerates insulation ageing; poor flow regulation disrupts the process.

Background and problem. A conventional speed controller meets the nominal settling-time requirement but frequently commands high transient current. Operators observe increased cabinet temperature and estimate that maintenance intervals are shortening. The plant therefore seeks a controller that maintains flow quality while reducing electrical stress.

Model and objectives. Around an operating point, the drive and process are represented by a linearized state model with states consisting of speed deviation, flow deviation, and a thermal state. The proposed cost is

\[

J=\int_0^T\left(q_f e_f^2+q_s e_s^2+r_i i^2+r_\theta \theta^2\right)dt,

\]

where \(e_f\) is flow error, \(e_s\) speed error, \(i\) current deviation, and \(\theta\) thermal deviation. The current is constrained by the inverter rating, and the thermal state must remain below a protection threshold.

Analysis and approach. An LQR design provides an initial gain after states are scaled into comparable engineering units. Several values of \(r_i\) are tested. Low current penalty gives fast regulation but produces near-saturation commands. A higher penalty reduces current peaks but increases flow settling time. Because the real actuator is bounded, the unconstrained LQR law is placed inside a constrained supervisory layer that limits current and applies anti-windup and rate protection. A finite-horizon optimization is also evaluated for scheduled demand changes.

The engineering team compares candidates using a Pareto plot containing integrated flow error, peak current, estimated energy, and thermal excursion. A candidate with the smallest quadratic cost is not automatically selected. The final choice must satisfy the hard thermal limit, preserve adequate flow response, and offer acceptable lifecycle cost. Verification includes nonlinear simulation, parameter variation, sensor bias tests, actuator saturation, and hardware-in-the-loop experiments.

Outcome. The illustrative study shows why objective formulation matters. If current is absent from the cost, the controller has no reason to conserve it except through indirect dynamic effects. If thermal protection is represented only by a small penalty, a mathematically attractive trajectory may still be unacceptable. The preferred design is therefore the feasible Pareto solution judged acceptable by operations, maintenance, and safety stakeholders.

Learning points. First, weights should represent measurable priorities and be subjected to sensitivity analysis. Second, constraints should be tested explicitly rather than assumed to emerge from optimization. Third, lifecycle and risk metrics can change the preferred controller even when transient plots appear similar.

9.8 Practical implementation and professional responsibilities

Optimal-control designs fail in practice for reasons that are often outside the Riccati equation. Common problems include poor state scaling, an inaccurate equilibrium model, neglected delays, sensor bias, actuator saturation, quantization, computational latency, and inappropriate noise assumptions. A controller should therefore be tested through a progression of models: nominal linear simulation, nonlinear simulation, uncertainty analysis, real-time simulation, and controlled commissioning.

Model validation is especially important because optimal control can exploit model features aggressively. An unmodeled resonance or neglected actuator lag may be excited by a controller that is optimal for the nominal model. Robustness margins, gain and phase analysis, Monte Carlo testing, and worst-case parameter studies complement the nominal optimality result.

Digital implementation introduces sampling and numerical issues. Continuous-time gains must be discretized consistently, and the sampling interval must resolve the fastest relevant dynamics. State estimators require attention to covariance tuning, outliers, sensor dropout, and initialization. Safety functions should remain independent enough to place the plant in a safe state if the optimization layer fails.

Ethically and professionally, the designer must make trade-offs visible. A reduced energy cost is not an adequate justification for increased probability of unsafe operation. Data used to estimate degradation or failure risk should be traceable, and assumptions should be communicated to decision-makers. Cybersecurity also matters when optimal policies depend on networked measurements or remotely updated models: corrupted measurements can produce systematically harmful control actions.

9.9 Current developments and emerging directions

Model predictive control (MPC) applies finite-horizon optimization repeatedly, using new measurements to update the decision. It is particularly useful when input and state constraints are central. Its success depends on prediction quality, computational capacity, horizon selection, terminal ingredients, and reliable fallback behavior.

Approximate dynamic programming and reinforcement learning seek policies when models are incomplete or high-dimensional. Their use in safety-critical control requires constraint handling, stability evidence, uncertainty quantification, interpretable validation, and carefully designed exploration. Data-driven policies should not be accepted merely because they perform well on a limited test set.

Risk-aware and distributionally robust optimization address uncertainty in disturbance distributions. Digital twins can support lifecycle evaluation by linking controller decisions to degradation and maintenance models. Cloud or edge analytics may support fleet-level tuning, while real-time embedded computation retains responsibility for fast safety-critical loops. These developments do not remove the need for physical modeling; they increase the importance of model governance, data quality, and verification.

Best-practice design workflow

A defensible optimal-control study generally follows this sequence:

1. Define the operational purpose, boundaries, stakeholders, and non-negotiable safety constraints.

2. Select and validate a model, including relevant disturbances, delays, actuator limits, and measurement characteristics.

3. Normalize states, inputs, and objectives so that weighting decisions have interpretable meaning.

4. Formulate the cost functional and distinguish hard constraints from soft preferences.

5. Select a method: LQR or LQG for suitable linear unconstrained problems, constrained predictive control for explicit limits, and nonlinear, robust, or stochastic methods when required by the application.

6. Perform sensitivity, robustness, and uncertainty analyses rather than reporting only nominal performance.

7. Compare alternatives using energy, effort, risk, reliability, and lifecycle metrics as appropriate.

8. Verify implementation through simulation, real-time testing, fault injection, and monitored commissioning.

9. Document assumptions, weights, failure modes, approval criteria, and procedures for revalidation after plant or software changes.

Chapter Summary

Optimal control formulates control design as the minimization of a performance functional subject to system dynamics, boundary conditions, and constraints. The calculus of variations and Pontryagin’s principle provide necessary trajectory conditions, while dynamic programming produces the value function and the Hamilton–Jacobi–Bellman equation. For linear systems with quadratic costs, these ideas lead to the Riccati equation and the LQR state-feedback law. When states are not directly available and disturbances and measurement noise are modeled as Gaussian, a Kalman filter combined with LQR produces the LQG controller under the separation principle.

A technically correct optimum is not necessarily an appropriate engineering decision. Control effort must be distinguished from physical energy, and average performance must be considered alongside constraint violation, uncertainty, risk, reliability, and lifecycle cost. Multi-objective and Pareto frameworks make trade-offs more transparent. Successful application requires model validation, constraint handling, robustness testing, safe digital implementation, and professional governance. Emerging MPC, approximate dynamic programming, reinforcement learning, and risk-aware methods expand the available tools but do not eliminate the need for defensible objectives and verification.

Key Terms

Optimal control: Selection of an admissible control policy that minimizes a specified performance functional.

Performance index: Mathematical measure used to quantify the quality or cost of a control trajectory.

Running cost: Cost accumulated continuously during system operation.

Terminal cost: Cost assigned to the state at the end of a finite horizon.

Admissible control: A control signal satisfying the system, regularity, input, and state requirements of the problem.

Costate: Auxiliary variable representing the marginal value of the system state in optimal-control analysis.

Hamiltonian: Function combining running cost and state dynamics through the costate.

Pontryagin’s minimum principle: Necessary conditions characterizing an optimal control trajectory.

Value function: Minimum future cost from a specified state and time.

Bellman’s principle: Principle that every remaining segment of an optimal policy is optimal for the state reached at that point.

HJB equation: Partial differential equation satisfied by the value function in dynamic programming.

LQR: Linear-quadratic regulator using state feedback to minimize a quadratic infinite- or finite-horizon cost.

CARE: Continuous-time algebraic Riccati equation used to compute the steady-state LQR solution.

LQG: Combination of LQR control and Kalman state estimation for linear stochastic systems.

Separation principle: Result allowing controller and estimator gains to be designed independently under standard LQG assumptions.

Control effort: Weighted measure of actuator activity in an objective function.

Pareto optimum: Solution for which improvement in one objective requires deterioration of another.

Lifecycle cost: Total discounted cost of acquisition, operation, maintenance, failure, and disposal.

Risk-sensitive control: Control design that explicitly penalizes uncertainty, variability, or adverse tail outcomes.

Model predictive control: Receding-horizon optimization that repeatedly computes control actions using updated measurements.

Review Questions

Short Answer Questions

1. What roles do the running cost and terminal cost play in a finite-horizon performance index?

2. State Bellman’s principle of optimality.

3. What is the purpose of the matrix \(R\) in an LQR problem?

4. Define the separation principle in LQG control.

5. Why should hard safety limits generally not be represented only as soft penalties?

Long Answer Questions

1. Derive the continuous-time LQR feedback law and the associated algebraic Riccati equation from a quadratic value function.

2. Explain the relationship between the calculus of variations, Pontryagin’s minimum principle, and dynamic programming.

3. Describe the LQG architecture, including the plant model, Kalman estimator, controller, and assumptions behind separation.

4. Discuss how energy, actuator effort, risk, and lifecycle cost can be incorporated into an engineering control decision.

5. Explain the main causes of mismatch between nominal optimal-control performance and field performance.

Analytical / Application Questions

1. A motor controller has excellent speed tracking but repeatedly reaches its current limit. Propose a revised objective and implementation strategy, and explain which requirements should be hard constraints.

2. Two controllers lie on a Pareto frontier: one has lower energy use but higher settling time and risk of flow violation. Develop a decision procedure for selecting between them.

3. An LQG controller performs well in simulation but becomes unsafe when a sensor develops a bias. Analyze why the nominal design is vulnerable and propose verification and mitigation measures.

22. Multiple Choice Questions with Answers

Question 1

Which expression represents a standard infinite-horizon continuous-time LQR cost?

A. J = max ||x(t)|| only

B. J = ∫₀∞ (xᵀQx + uᵀRu) dt

C. J = ∫₀∞ (xᵀRx + uᵀQu) dt with no restrictions

D. J = Φ(x(T)) for T = ∞ only

Correct Answer: B. J = ∫₀∞ (xᵀQx + uᵀRu) dt

Question 2

In dynamic programming, the value function V(x,t) represents:

A. The measured output at time t

B. The actuator saturation limit

C. The minimum future cost from state x at time t

D. The covariance of measurement noise

Correct Answer: C. The minimum future cost from state x at time t

Question 3

For the continuous-time LQR problem, the optimal state-feedback law is:

A. u = R⁻¹BᵀPx

B. u = -R⁻¹BᵀPx

C. u = -Q⁻¹Aᵀx

D. u = Px only

Correct Answer: B. u = -R⁻¹BᵀPx

Question 4

What equation determines the steady-state LQR matrix P?

A. The Euler integration equation

B. The measurement covariance equation only

C. The continuous-time algebraic Riccati equation

D. The characteristic equation of R alone

Correct Answer: C. The continuous-time algebraic Riccati equation

Question 5

Increasing the control-effort weighting R will generally tend to:

A. Penalize actuator activity more strongly

B. Remove all state penalties

C. Guarantee zero settling time

D. Make measurement noise Gaussian

Correct Answer: A. Penalize actuator activity more strongly

Question 6

The LQG controller uses which estimate in its feedback law?

A. The terminal cost only

B. The estimated state produced by a Kalman filter

C. The control input covariance only

D. The open-loop initial state at all times

Correct Answer: B. The estimated state produced by a Kalman filter

Question 7

Which statement best describes a Pareto-optimal solution?

A. It minimizes every objective independently

B. It is always the solution with the smallest energy

C. No objective can be improved without worsening at least one other objective

D. It ignores all constraints

Correct Answer: C. No objective can be improved without worsening at least one other objective

Question 8

Why can an unconstrained LQR solution be unsuitable for an actual actuator?

A. LQR cannot use a state model

B. The computed control may exceed physical amplitude or rate limits

C. The Riccati equation contains no matrices

D. LQR is defined only for nonlinear systems

Correct Answer: B. The computed control may exceed physical amplitude or rate limits

Question 9

In the illustrative industrial-drive case, current was included in the cost primarily to:

A. Increase sensor noise

B. Represent actuator stress and electrical burden

C. Eliminate the process model

D. Guarantee that flow error is zero instantly

Correct Answer: B. Represent actuator stress and electrical burden

Question 10

Which method is especially suited to repeatedly optimizing while explicitly handling input and state constraints?

A. Model predictive control

B. Unconstrained pole placement only

C. Static open-loop scheduling only

D. A terminal-cost calculation without dynamics

Correct Answer: A. Model predictive control