6.5 Control Objective

A control objective is the specification of what a control system is designed to achieve: the desired relationship between the system's outputs and a set of reference signals and performance criteria. It defines, in precise terms, what constitutes success in controlling the system and thereby determines the criterion against which any proposed control design is evaluated. A well-formulated control objective translates the user's operational goals into mathematical requirements that can guide controller design and performance verification.

Control objectives fall into several fundamental categories. Regulation objectives specify that the controlled output should remain close to a fixed set point despite the presence of disturbances. Tracking objectives specify that the output should follow a time-varying reference signal with acceptable fidelity and speed. Stabilization objectives specify that the output of an inherently unstable system should be prevented from diverging. Optimization objectives specify that a performance index measuring some notion of quality should be maximized or minimized. Most real control problems combine elements of several of these categories simultaneously.

The translation of operational goals into formal control objectives requires the specification of performance metrics that quantify the desired behavior. For step reference changes, the standard time-domain metrics define the control objective in terms of the transient and steady-state response:

Rise time (T_r): the time for the output to rise from 10% to 90% of the step amplitude, specifying how quickly the system responds.

Overshoot (M_p): the maximum percentage by which the output exceeds the final set point, specifying how much the response oscillates.

Settling time (T_s): the time after which the output remains within a specified tolerance band (typically ±2% or ±5%) of the set point, specifying when the response is effectively complete.

Steady-state error (e_ss): the residual offset between the output and the set point after the transient has ended, specifying the accuracy of the regulation.

These metrics can be related to the closed-loop pole locations of the system. For a second-order closed-loop system with undamped natural frequency ω_n and damping ratio ζ, the percent overshoot is:

This relationship shows that specifying a maximum acceptable overshoot directly implies a lower bound on the required damping ratio ζ. Similarly, specifying a maximum settling time implies a lower bound on the product ζω_n, the real part of the closed-loop poles. The control objective in terms of time-domain metrics thus translates into a region in the complex s-plane within which the closed-loop poles must lie.

Frequency-domain control objectives specify performance requirements in terms of the system's gain and phase characteristics across frequency. A bandwidth requirement specifies the range of frequencies over which the system must track references accurately, stated as the frequency at which the closed-loop magnitude response drops to −3 dB of its low-frequency value. A disturbance rejection requirement specifies how much the sensitivity function must be attenuated at specific frequencies where disturbances are concentrated. A noise rejection requirement specifies that the complementary sensitivity function must be small at high frequencies where measurement noise dominates. These frequency-domain objectives directly constrain the loop transfer function design through the Bode and Nyquist frameworks.

Robust control objectives add a further dimension by specifying that the desired performance must be maintained not just at the nominal plant model but across a range of model uncertainty. An H∞ control objective minimizes the infinity norm of the closed-loop transfer function from disturbances to regulated outputs, which corresponds to minimizing the worst-case gain across all frequencies. This objective explicitly accounts for model uncertainty by requiring good performance under the worst-case perturbation within a specified uncertainty set, producing a controller that is inherently conservative but guaranteed to meet performance requirements across the uncertainty range.

Optimal control objectives define performance in terms of a cost functional that integrates a measure of control effort and regulation error over time. The linear quadratic regulator (LQR) minimizes:

where Q is a positive semidefinite matrix weighting state deviations and R is a positive definite matrix weighting control effort. The matrices Q and R encode the control objective: choosing Q large relative to R emphasizes tight regulation with expensive control action; choosing R large relative to Q emphasizes economical use of control authority at the cost of allowing larger state deviations. This formulation makes the tradeoff between regulation quality and control cost explicit and allows it to be systematically explored.

In biological systems, control objectives are not explicitly programmed but emerge from evolutionary selection pressures. The objective implicit in the baroreflex is to maintain arterial blood pressure within a range compatible with adequate organ perfusion across a wide range of postures, activities, and vascular conditions. The objective implicit in motor control is to achieve accurate, coordinated movements with minimum metabolic cost and minimum risk of injury. These biological control objectives are reflected in the structure, gains, and dynamics of the biological regulatory systems that evolution has refined.

In organizational management, control objectives are expressed as strategic goals, performance targets, and operational standards. A production control objective specifies the output rate, quality level, and cost per unit to be maintained. A financial control objective specifies target profitability ratios, liquidity levels, and capital adequacy metrics. A safety regulatory objective specifies maximum acceptable accident rates, exposure limits, or failure probabilities. The design of organizational control mechanisms—reporting structures, performance metrics, review cycles, and incentive systems—is driven by these objectives, and the evaluation of organizational performance consists of assessing how closely the realized outcomes match the stated control objectives.

Control objectives must be internally consistent and physically achievable. An objective that simultaneously demands zero steady-state error, zero overshoot, instantaneous response, and complete disturbance rejection cannot be satisfied by any finite-bandwidth control system operating under physical constraints. The specification of control objectives must therefore account for the fundamental limitations imposed by causality, the plant's dynamics, actuator constraints, measurement noise, and plant uncertainty. The disciplined formulation of realistic, achievable, and prioritized control objectives is the essential first step in any systematic control system design, and the quality of the final control design is ultimately bounded by the accuracy and consistency of the objectives against which it was designed.