Optimal Control and Model Predictive Control

Introduction and Context

Classical control design often begins with a prescribed objective such as tracking a reference, rejecting disturbances, or maintaining stability. Optimal control extends this viewpoint by treating control design as a decision problem: among all admissible inputs, which one gives the best compromise between performance, energy, actuator activity, risk, and computational effort?

This question is central in aerospace guidance, autonomous vehicles, industrial process control, robotics, power electronics, and communication systems. A controller that minimizes tracking error alone may demand excessive actuator power. A controller that minimizes energy alone may respond too slowly. A practical design therefore requires an explicit performance index and constraints representing the physical system.

Optimal control provides a systematic framework for such decisions. Calculus of variations and Pontryagin’s minimum principle provide necessary conditions for continuous-time solutions. Dynamic programming gives a recursive viewpoint and leads to the Hamilton–Jacobi–Bellman equation. Linear-quadratic regulation (LQR) provides an important closed-form solution for linear systems with quadratic costs. Linear-quadratic-Gaussian (LQG) control combines LQR with state estimation. Model predictive control (MPC) repeatedly solves a finite-horizon optimization problem while explicitly handling constraints.

The central engineering challenge is not simply to obtain a mathematically optimal input. It is to obtain a solution that is stable, robust to model uncertainty, implementable in real time, and consistent with safety and operational requirements.

Learning Objectives

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

1. Formulate control objectives as constrained optimization problems.

2. Explain the roles of calculus of variations, optimality conditions, and dynamic programming.

3. Derive and interpret the continuous-time LQR solution through the algebraic Riccati equation.

4. Distinguish LQR, LQG, and MPC and identify when each is appropriate.

5. Construct a basic finite-horizon MPC problem and explain receding-horizon implementation.

6. Evaluate trade-offs among tracking quality, energy, robustness, risk, and computation.

7. Identify practical, ethical, and governance requirements for safety-critical optimal control.

1. Formulating an Engineering Decision as an Optimization Problem

Consider a continuous-time dynamical system

\[

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

\]

where \(x\in\mathbb{R}^n\) is the state, \(u\in\mathbb{R}^m\) is the control input, and \(y\) is the measured output. An optimal-control problem selects an admissible input trajectory \(u(t)\) to minimize a cost functional

\[

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

\]

Here, \(L\) is the running cost, \(\Phi\) is the terminal cost, and \(T\) is the final time. Typical terms include tracking error, control energy, state excursions, actuator wear, and economic operating cost. For example,

\[

L=(x-x_r)^TQ(x-x_r)+u^TRu,

\]

where \(x_r\) is a desired state, \(Q\succeq0\) weights state error, and \(R\succ0\) weights control effort. The notation \(Q\succeq0\) means that \(Q\) is positive semidefinite, while \(R\succ0\) means that \(R\) is positive definite.

The optimization is subject to system dynamics and constraints such as

\[

x(t)\in\mathcal{X},\qquad u(t)\in\mathcal{U},\qquad g(x(t),u(t))\leq0.

\]

These constraints may represent voltage limits, current ratings, actuator saturation, temperature bounds, collision avoidance, or communication bandwidth. They are not merely mathematical details: omitting them can produce a controller that is optimal in simulation but unsafe in operation.

A useful cost design is dimensionally consistent. If position is measured in metres and force in newtons, the weights must convert their squared quantities to a common cost scale. Weight selection is therefore a statement of engineering priorities, not a purely numerical tuning exercise.

2. Variational and Hamiltonian Foundations

2.1 Calculus of variations

In calculus of variations, the decision variable is a function rather than a finite-dimensional vector. For a functional

\[

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

\]

an unconstrained stationary trajectory satisfies the Euler–Lagrange equation

\[

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

\]

In optimal control, the state derivative is constrained by the system dynamics, so a Lagrange multiplier, commonly called the costate \(\lambda(t)\), is introduced. Define the Hamiltonian

\[

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

\]

For an unconstrained control and appropriate smoothness assumptions, necessary conditions include

\[

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

\]

\[

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

\]

and

\[

\frac{\partial H}{\partial u}=0.

\]

The state has an initial condition, while the costate generally has a terminal condition determined by the terminal cost. If the terminal state is free and \(\Phi\) is differentiable,

\[

\lambda(T)=\frac{\partial\Phi}{\partial x}(x(T),T).

\]

These are necessary, not generally sufficient, conditions. A stationary solution may be a maximum, a saddle point, or a local minimum. Control constraints replace the stationarity condition with a minimization or complementary-slackness condition, as formalized by Pontryagin’s minimum principle.

2.2 Dynamic programming

Dynamic programming defines the optimal cost-to-go or 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].

\]

The Hamilton–Jacobi–Bellman equation is

\[

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

\]

with terminal condition \(V(x,T)=\Phi(x,T)\). The minimizing input is obtained from the expression inside braces.

Dynamic programming is conceptually powerful because it expresses optimality recursively: an optimal policy consists of an optimal first decision followed by an optimal policy for the remaining problem. However, solving the HJB equation becomes computationally prohibitive as state dimension increases. This difficulty is known as the curse of dimensionality. LQR and MPC can be viewed as practical specializations that avoid solving a general high-dimensional HJB equation.

3. Linear-Quadratic Regulation

3.1 Problem definition

For the linear time-invariant system

\[

\dot{x}=Ax+Bu,

\]

consider the infinite-horizon cost

\[

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

\]

where \(Q\succeq0\) and \(R\succ0\). The objective is regulation toward the origin. Tracking a nonzero constant reference requires a coordinate transformation, integral augmentation, feedforward term, or a more general tracking formulation.

Assume that \((A,B)\) is stabilizable and \((Q^{1/2},A)\) is detectable. Under standard additional conditions, the continuous-time algebraic Riccati equation (CARE)

\[

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

\]

has a unique positive semidefinite stabilizing solution \(P\). The optimal state-feedback law is

\[

u^*(t)=-Kx(t),\qquad K=R^{-1}B^TP.

\]

The closed-loop matrix is \(A-BK\). The stabilizing solution is the one for which this matrix is Hurwitz, meaning all eigenvalues have negative real parts.

3.2 Interpretation of the weights

Increasing \(Q\) relative to \(R\) penalizes state deviation more strongly and usually produces faster, more aggressive control. Increasing \(R\) penalizes input effort and generally produces slower, less energetic responses. These statements are qualitative: the exact effect depends on the system dynamics and matrix structure.

The cost has units only after the weights are assigned suitable units. A practical normalization is to scale each state by an acceptable maximum deviation and each input by its allowable range, then tune dimensionless weights. This reduces numerical conditioning problems and makes weights easier to interpret.

3.3 Finite-horizon LQR and the Riccati differential equation

For a finite horizon \([0,T]\) with terminal cost \(x(T)^TS_fx(T)\), the optimal value has the form

\[

V(x,t)=x^TP(t)x,

\]

where \(P(t)\) satisfies the Riccati differential equation

\[

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

\qquad P(T)=S_f.

\]

The time-varying optimal gain is

\[

K(t)=R^{-1}B^TP(t).

\]

The infinite-horizon CARE is the steady-state form of this equation under appropriate convergence conditions.

3.4 A scalar illustration

Consider

\[

\dot{x}=u,

\qquad J=\int_0^\infty (x^2+u^2)dt.

\]

Here \(A=0\), \(B=1\), \(Q=1\), and \(R=1\). Let \(P=p\). The CARE becomes

\[

0=0+0-p^2+1,

\]

so the nonnegative solution is \(p=1\). Therefore \(K=1\), and

\[

u^*=-x.

\]

The closed-loop system is \(\dot{x}=-x\), which is asymptotically stable. The example demonstrates the balance between state error and control effort: a larger input penalty would reduce the gain, while a larger state penalty would increase it.

4. LQG Control: Optimal Control with Noisy Measurements

LQR assumes that the complete state is available. In many systems only an output is measured:

\[

y=Cx+v,

\]

where \(v\) is measurement noise. The state may be estimated using a Kalman filter based on the stochastic model

\[

\dot{x}=Ax+Bu+w,

\qquad y=Cx+v,

\]

where \(w\) and \(v\) are commonly modeled as zero-mean white Gaussian processes with covariance matrices \(W\) and \(V\), respectively. The Kalman filter computes an estimate \(\hat{x}\); the LQG control law is

\[

u=-K\hat{x},

\]

where \(K\) is the LQR gain.

The separation principle states, under standard linear-Gaussian assumptions and appropriate stabilizability/detectability conditions, that the controller and estimator can be designed independently. This does not mean the combined system is insensitive to modeling errors. Poor noise covariance assumptions, unmodeled dynamics, sensor bias, saturation, and nonlinearities can substantially degrade performance.

LQG minimizes an expected quadratic cost for the assumed stochastic model. It does not directly guarantee robust performance for arbitrary disturbances. In safety-critical systems, Kalman filtering and LQR are often supplemented by robust design, fault detection, constraint handling, and supervisory protection.

5. Model Predictive Control

5.1 Basic principle

MPC uses a model to predict future behavior over a finite prediction horizon. At each sampling time it solves an optimization problem, applies only the first input or first input interval, measures the new state, shifts the horizon forward, and repeats. This is called receding-horizon control.

For a discrete-time linear model,

\[

x_{k+1}=Ax_k+Bu_k,

\]

a typical quadratic MPC problem is

\[

\min_{u_{k|k},\ldots,u_{k+N-1|k}}

\sum_{i=0}^{N-1}\left[(x_{k+i|k}-r_{k+i})^TQ(x_{k+i|k}-r_{k+i})+u_{k+i|k}^TRu_{k+i|k}\right]

+ x_{k+N|k}^TP_fx_{k+N|k},

\]

subject to

\[

x_{k+i+1|k}=Ax_{k+i|k}+Bu_{k+i|k},

\]

\[

x_{k+i|k}\in\mathcal{X},\qquad u_{k+i|k}\in\mathcal{U}.

\]

The initial condition is \(x_{k|k}=\hat{x}_k\), the current measured or estimated state. The subscript \(i|k\) means a prediction for time \(k+i\) made at time \(k\).

5.2 Why constraints matter

MPC’s major practical advantage over unconstrained LQR is explicit constraint handling. For example, a motor-drive controller may require

\[

|u_k|\leq u_{\max},\qquad |i_k|\leq i_{\max},

\]

while a vehicle may require steering, speed, and collision-avoidance constraints. Infeasibility can occur when the current state cannot be driven into the admissible region within the chosen horizon. Designs therefore commonly include a terminal invariant set, soft constraints with carefully penalized slack variables, or a supervisory fallback controller.

A soft state constraint can be represented using a slack variable \(\epsilon\geq0\):

\[

x\leq x_{\max}+\epsilon,

\]

with an added penalty such as \(\rho\epsilon^2\) or \(\rho\epsilon\). Softening should not be used to conceal dangerous violations; the slack value must be monitored and associated with an appropriate safety response.

5.3 Stability and recursive feasibility

Applying a finite-horizon optimum does not automatically guarantee closed-loop stability. Common sufficient design ingredients include a terminal cost based on an LQR Riccati solution and a terminal set that is positively invariant under a local controller. Recursive feasibility means that if the optimization is feasible now, feasibility can be maintained at later sampling times, subject to assumptions.

Robust MPC addresses disturbances and uncertainty by tightening constraints, using tube-based predictions, min-max optimization, or disturbance-feedback policies. These methods generally increase conservatism and computational cost. A design must balance nominal performance against the risk of constraint violation under model mismatch.

5.4 Computational structure

After eliminating predicted states, linear-quadratic MPC commonly becomes a quadratic program (QP):

\[

\min_z \frac12z^THz+f^Tz

\quad\text{subject to}\quad Gz\leq b,

\]

where \(z\) collects future control moves. If the model, horizon, and constraints are fixed, much of the problem structure can be precomputed. Sparse QP solvers, warm starts, code generation, and explicit MPC can support real-time implementation. Nonlinear MPC leads to nonlinear programs and requires greater computational effort, reliable initialization, and careful treatment of local minima.

6. Comparing LQR, LQG, and MPC

LQR is attractive when the model is approximately linear, the operating region is unconstrained, and full-state feedback is available or reliably estimated. It provides a transparent gain and strong analytical results at low computational cost.

LQG adds stochastic state estimation and is appropriate when Gaussian noise models are credible. It remains computationally efficient but does not naturally enforce hard actuator or state constraints.

MPC is preferable when constraints, multivariable interactions, preview information, or changing references are central. Its disadvantages include online computation, tuning complexity, dependence on model quality, and possible infeasibility. Nonlinear MPC can represent a wider range of physics but increases these challenges.

A useful selection principle is to use the simplest method that satisfies the operational requirements. Adding optimization sophistication without reliable sensing, a suitable model, or a validated fallback strategy can reduce rather than improve system safety.

7. A Design Workflow for Engineering Practice

A disciplined workflow is as follows.

Define the decision and risk. Specify the controlled variables, manipulated variables, reference trajectories, disturbances, failure modes, and unacceptable outcomes.

Develop and validate the model. Determine the operating region, sampling time, actuator dynamics, sensor characteristics, delays, and uncertainty. Validate against experiments or high-fidelity simulation.

Select the objective. Separate tracking, energy, smoothness, economic cost, and risk terms. Normalize variables and document the rationale for weights.

Represent constraints explicitly. Include physical limits, rate limits, safety envelopes, and operating regulations. Distinguish hard constraints from constraints that may be softened under a controlled emergency policy.

Choose the architecture. Select LQR/LQG for suitable unconstrained linear problems, MPC for constrained multivariable operation, and robust or nonlinear variants when uncertainty or nonlinear behavior justifies them.

Analyze stability and feasibility. Examine closed-loop poles, terminal conditions, constraint margins, disturbance rejection, estimator behavior, and worst-case scenarios.

Test progressively. Use software-in-the-loop, processor-in-the-loop, hardware-in-the-loop, and controlled experiments. Include sensor faults, actuator saturation, communication delays, and model mismatch.

Monitor online. Record constraint margins, optimization status, residuals, estimator innovations, computational time, and fallback activation. A controller should expose evidence that it is operating within its validated envelope.

8. Illustrative Case: Constrained Temperature Control of a Multizone Chamber

The following is a fictional but representative engineering case.

Background

A laboratory thermal chamber has two interacting zones. Electric heaters influence both zones, while heat exchange between zones creates coupling. The objective is to track temperature references while limiting heater power and avoiding excessive temperature rate.

Problem

A decentralized proportional controller performs adequately for constant references but overshoots when both references change. Heater saturation causes slow recovery, and independent controllers ignore thermal coupling. Energy consumption is also higher than necessary.

A discrete-time linear model is identified around the normal operating point:

\[

x_{k+1}=Ax_k+Bu_k,

\]

where \(x_k\) contains temperature deviations in the two zones and \(u_k\) contains heater-power deviations. The constraints are

\[

0\leq u_{j,k}\leq 100\%,

\]

and temperature deviations must remain within a specified operating band. The MPC objective penalizes temperature tracking error, heater power, and changes in heater power:

\[

J=\sum_{i=0}^{N-1}\left(\|x_{k+i|k}-r\|_Q^2+\|u_{k+i|k}\|_R^2+\|\Delta u_{k+i|k}\|_S^2\right)+\|x_{k+N|k}-r\|_{P_f}^2.

\]

Here \(\|z\|_Q^2=z^TQz\), and \(\Delta u_k=u_k-u_{k-1}\). The rate term reduces abrupt heater switching and can extend actuator life.

Analysis and approach

The model is discretized at a sampling interval short enough to capture the dominant thermal dynamics. State estimation is used because only temperatures are measured and sensor noise is present. The horizon is selected to cover the principal thermal response, while the solver is benchmarked against the available processor time.

A terminal cost derived from an unconstrained LQR design improves end-of-horizon behavior. Heater limits are hard constraints. A small, heavily penalized slack variable is permitted only for a noncritical comfort band; the absolute safety temperature limit remains hard. If the QP fails or exceeds a computation-time limit, a validated fallback controller holds the last safe command or applies a bounded stabilizing law.

Outcome

Compared with decentralized proportional control, the MPC design is expected to reduce overshoot, coordinate the heaters, and reduce unnecessary heater activity. The primary engineering result is not merely a lower nominal cost. It is improved behavior near saturation and transparent enforcement of operating limits. Before deployment, the design must be tested under sensor bias, heater failure, unmodeled heat loads, and communication loss.

Learning points

The case illustrates that MPC is valuable because it combines prediction, multivariable coordination, and constraint management. It also shows that a cost function cannot replace a safety architecture: hard limits, fault handling, validation, and fallback control remain necessary.

9. Limitations, Governance, and Contemporary Developments

Optimality is always relative to a model and a cost function. An inaccurate model can produce systematically poor decisions. Weight tuning can hide value judgments, such as trading energy consumption against comfort or production rate against equipment life. In safety-critical applications, these trade-offs must be documented and approved rather than left to informal experimentation.

Data-driven and learning-based control can improve model prediction, adapt to changing dynamics, or accelerate optimization. However, learned models may extrapolate poorly, and an AI-based controller may be difficult to certify. A responsible architecture places learning inside a monitored envelope, uses uncertainty estimates where possible, and retains interpretable safety constraints and a verified fallback policy.

Cybersecurity is also a control concern. Manipulated measurements, delayed commands, or denial of service can make an optimizer select unsafe actions. Authentication, secure logging, anomaly detection, network segmentation, and fail-safe behavior should be considered during design.

Best practices include version-controlled models and parameters, traceable requirements, independent verification, documented solver tolerances, deterministic timing analysis, constraint-margin monitoring, and post-deployment review. Regulatory obligations vary by sector, but the general principle is consistent: the organization deploying an optimal controller remains responsible for its decisions and operational consequences.

Chapter Summary

Optimal control formulates control design as the minimization of a cost functional subject to system dynamics and physical constraints. Calculus of variations and Hamiltonian methods produce necessary state–costate conditions, while dynamic programming expresses optimality through the value function and the HJB equation. For linear systems with quadratic costs, LQR yields a state-feedback law from the Riccati equation. LQG combines LQR with Kalman state estimation under linear-Gaussian assumptions. MPC repeatedly solves a finite-horizon constrained optimization problem, applies the first control action, and uses new measurements to update the plan. Its principal advantages are prediction, multivariable coordination, and explicit constraint handling; its challenges include computational demand, model uncertainty, infeasibility, and tuning. Professional implementation requires validated models, dimensional consistency, stability and feasibility analysis, robust testing, monitoring, cybersecurity, and a safe fallback strategy.

Key Terms

Optimal control: Selection of an input trajectory that minimizes a specified cost subject to dynamics and constraints.

Running cost: Instantaneous contribution to an optimal-control objective.

Terminal cost: Cost assigned to the state at the final time or horizon endpoint.

Costate: Lagrange-multiplier trajectory associated with system dynamics.

Hamiltonian: Function combining running cost and dynamics weighted by the costate.

Calculus of variations: Mathematical study of extrema of functionals.

Dynamic programming: Recursive decomposition of an optimization problem into cost-to-go decisions.

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

LQR: Linear-quadratic regulator for linear dynamics and quadratic cost.

CARE: Continuous-time algebraic Riccati equation.

Riccati equation: Matrix differential or algebraic equation determining optimal quadratic-control gains.

LQG: Linear-quadratic-Gaussian control combining LQR and Kalman estimation.

Kalman filter: Optimal linear state estimator under standard Gaussian assumptions.

MPC: Receding-horizon control based on repeated finite-horizon optimization.

Prediction horizon: Number of future sampling steps considered by MPC.

Terminal set: Set used to support feasibility or stability at the end of an MPC horizon.

Recursive feasibility: Property that feasibility can be maintained at successive control steps.

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

Robust MPC: MPC designed to account explicitly for disturbances and model uncertainty.

Receding horizon: Strategy of applying only the first optimized action before resolving the problem.

Review Questions

Short Answer Questions

1. What are the roles of the running cost and terminal cost in an optimal-control problem?

2. State the continuous-time algebraic Riccati equation for LQR.

3. Why does MPC apply only the first element of an optimized input sequence?

4. What is the separation principle in LQG control?

5. Distinguish a hard constraint from a soft constraint.

Long Answer Questions

1. Explain how calculus of variations and the Hamiltonian lead to state, costate, and stationarity conditions.

2. Derive and interpret the continuous-time LQR feedback law and discuss the assumptions required for a stabilizing solution.

3. Compare LQR, LQG, and MPC in terms of models, measurements, constraints, computational effort, and robustness.

4. Explain recursive feasibility and terminal ingredients in constrained MPC.

5. Discuss how cost-function design can balance tracking performance, energy use, actuator wear, and operational risk.

Analytical / Application Questions

1. For \(\dot{x}=u\) and \(J=\int_0^\infty(x^2+4u^2)dt\), formulate the CARE and determine the stabilizing scalar feedback gain.

2. Design a conceptual MPC formulation for a vehicle subject to speed, steering, and obstacle-avoidance constraints. Identify the state, input, objective terms, and fallback strategy.

3. A plant’s MPC becomes infeasible whenever a large disturbance occurs. Analyze at least three possible causes and propose technical remedies while preserving safety.

Multiple Choice Questions with Answers

Question 1

Which condition is associated with the continuous-time LQR problem?

A. The input cost matrix R is positive definite

B. The input cost matrix R is negative definite

C. The system must have no input constraints

D. The state weighting matrix Q must be nonsymmetric

Correct Answer: A. The input cost matrix R is positive definite

Question 2

For the scalar system ẋ = u with Q = 1 and R = 1 in the infinite-horizon LQR problem, what is the optimal feedback law?

A. u = x

B. u = -x

C. u = -2x

D. u = 0

Correct Answer: B. u = -x

Question 3

What does the costate represent in the Hamiltonian formulation?

A. A measurement-noise covariance

B. A Lagrange multiplier associated with system dynamics

C. The terminal control input

D. The sampling interval

Correct Answer: B. A Lagrange multiplier associated with system dynamics

Question 4

The Hamilton–Jacobi–Bellman equation is based primarily on which principle?

A. Superposition

B. Dynamic programming

C. Nyquist stability

D. Pole-zero cancellation

Correct Answer: B. Dynamic programming

Question 5

What is the defining operational feature of receding-horizon MPC?

A. It applies the entire optimized input sequence open loop

B. It applies only the first optimized action and then resolves the problem

C. It never uses a model

D. It requires an unconstrained system

Correct Answer: B. It applies only the first optimized action and then resolves the problem

Question 6

Which method directly handles actuator and state constraints in its standard formulation?

A. Unconstrained LQR

B. MPC

C. A fixed Kalman filter

D. Open-loop feedforward control

Correct Answer: B. MPC

Question 7

In LQG control, the Kalman filter is used to estimate which quantity?

A. The future reference only

B. The unmeasured system state

C. The Riccati matrix only

D. The actuator rating

Correct Answer: B. The unmeasured system state

Question 8

What is the main purpose of a terminal cost in finite-horizon MPC?

A. To represent end-of-horizon behavior and support stability

B. To eliminate all measurement noise

C. To make the model nonlinear

D. To remove input constraints

Correct Answer: A. To represent end-of-horizon behavior and support stability

Question 9

What does recursive feasibility mean in MPC?

A. The optimizer always has zero computational cost

B. Feasibility can be maintained at successive control steps under stated assumptions

C. The system has no disturbances

D. The prediction horizon is infinite

Correct Answer: B. Feasibility can be maintained at successive control steps under stated assumptions

Question 10

If the input penalty R is increased in a standard LQR design, what qualitative effect is generally expected?

A. More aggressive control effort

B. Less penalty on actuator use

C. A tendency toward less aggressive control

D. Automatic removal of all state constraints

Correct Answer: C. A tendency toward less aggressive control