Optimal Control within Guidance, Navigation, Estimation, and Control

Optimal control can appear disconnected from classical feedback control. Classical control courses often emphasize transfer functions, compensators, stability margins, and feedback loops. Optimal control, by contrast, introduces calculus of variations, necessary conditions, boundary-value problems, direct transcription, and numerical optimization.

The purpose of this session is to place optimal control within the larger architecture of a complete control system. The central conclusion is:

The resulting reference signals are then tracked by an inner-loop feedback controller using estimated states obtained from navigation and estimation.

Major Components of a Complete Control System¶

A practical closed-loop system commonly contains five major components:

1. Plant or dynamics: the physical system being controlled.

2. Measurement system: sensors that produce measurable outputs.

3. State estimation: an algorithm that reconstructs the system state.

4. Control: a feedback law that computes corrective control action.

5. Reference generation or guidance: a method for generating desired state and control trajectories.

These components are related but not interchangeable.

Plant Dynamics¶

The plant is described by a state-space model

The control input is $\boldsymbol{u}(t)$. The derivative $\dot{\boldsymbol{x}}(t)$ is integrated to obtain the state:

Thus, the plant block maps the state and control to the state derivative.

Measurement and Navigation¶

In general, the full state cannot be measured directly. Instead, sensors produce

where $\boldsymbol{v}(t)$ represents measurement noise.

In aerospace applications, the process of determining position, velocity, attitude, and related states is called navigation.

Navigation is fundamentally a measurement and state-determination problem.

Guidance¶

Guidance determines the desired path or trajectory.

A navigation system may determine that a vehicle is at a given latitude and longitude. A guidance system determines the desired route from that location to the destination.

This distinction is often blurred in everyday language. Turn-by-turn route instructions are guidance, even if they are marketed as navigation.

Control¶

Control generates the actuator commands required to track the reference trajectory.

A feedback controller typically maps tracking error into a corrective control:

The tracking error may be defined as

The full control is

Thus, guidance supplies the nominal control and state, while feedback control supplies corrections.

State Estimation¶

Because the state is not generally measured directly, an estimator combines the measured output and known control:

Examples include:

• Luenberger observers,

• Kalman filters,

• extended Kalman filters,

• unscented Kalman filters,

• particle filters.

The estimator output $\hat{\boldsymbol{x}}$ is an estimate, not the exact state.

Complete Guidance–Navigation–Estimation–Control Architecture¶

Reference-State and Reference-Control Generation¶

Guidance typically produces two signals:

The reference state describes the desired trajectory. The reference control is the nominal input required to follow it.

If the system is exactly on the reference trajectory,

then

Ideally,

and the plant is driven only by the nominal control $\boldsymbol{u}_{\mathrm{ref}}$.

Inner Loop and Outer Loop¶

The architecture naturally separates into two loops.

Inner Loop¶

The inner loop is the feedback controller:

It operates at a relatively high update rate and rejects disturbances, compensates for model error, and stabilizes the plant.

Outer Loop¶

The outer loop is guidance:

It is responsible for replanning and reference generation. It may operate less frequently because solving a guidance problem is often computationally expensive.

Optimal Control as Optimal Guidance¶

An optimal control problem generates a state and control trajectory that optimize a performance index:

subject to

The solution is

Within the full control architecture, these are interpreted as

Therefore,

In aerospace terminology,

Open-Loop Guidance¶

Suppose the optimal control problem is solved once before execution:

If this reference is never recomputed, the guidance solution is open loop.

The inner-loop controller may still be closed loop, but the guidance trajectory itself is fixed.

This distinction is important:

• open-loop optimal control does not imply that the entire vehicle is operated without feedback;

• it only means that the reference trajectory is not regenerated using new state information.

Receding-Horizon or Replanning Guidance¶

Suppose guidance is recomputed every $T_g$ seconds. At update time $t_k$,

The current estimated state becomes the new initial condition:

A new optimal control problem is solved from $t_k$ to the terminal time.

This produces updated reference signals:

This procedure is closely related to:

• model predictive control,

• receding-horizon control,

• trajectory replanning,

• midcourse correction.

Guidance Update Switch¶

Guidance can be viewed as operating through a switch that closes every $T_g$ seconds.

When the switch is open, the system continues to use the most recently generated reference.

When the switch closes:

1. the latest state estimate is passed to guidance;

2. the reference-generation problem is resolved;

3. new state and control references are issued.

The update period must be compatible with the computational time required to solve the guidance problem.

Computational Feasibility¶

A reference-generation method is useful only if it can provide a solution before the next guidance update.

If the guidance update period is $T_g$ and the solution time is $T_s$, practical implementation requires

For fast systems, $T_g$ may be very small. A highly accurate but slow optimization method may therefore be unusable in real time.

This motivates:

• simplified dynamics,

• reduced-order models,

• precomputed policies,

• lookup tables,

• explicit guidance laws,

• warm-started optimization,

• surrogate models,

• approximate optimal control.

Efficient Guidance versus Exact Optimal Guidance¶

A practical guidance law need not be exactly optimal.

A useful design may satisfy

while requiring substantially less computational effort.

Thus, practical engineering often seeks efficient guidance rather than mathematically exact optimal guidance.

Launch-Vehicle Ascent¶

Launch-vehicle ascent is a classic example of reference generation.

A typical ascent contains several phases:

1. vertical rise,

2. pitch-over maneuver,

3. gravity turn,

4. stage separation,

5. upper-stage burns,

6. orbital insertion.

Within each phase, the steering profile may be approximated using simple parameterizations, such as:

• constant pitch rate,

• piecewise-linear pitch history,

• low-order polynomial steering,

• linear-tangent steering.

These simple guidance laws can be extremely close to optimal because the physical structure of the problem is well understood.

Direct Shooting in Ascent Guidance¶

Suppose the control is represented by $N$ parameters:

The ascent optimization becomes

subject to the integrated vehicle dynamics and terminal orbital constraints.

This is direct shooting.

Direct shooting is especially attractive when the steering law can be accurately represented with only a few parameters.

Powered Explicit Guidance¶

Powered explicit guidance uses the current state and desired terminal state to construct a required velocity change.

Define

A simple guidance principle is to orient thrust approximately in the direction of $\Delta\boldsymbol{v}$:

The algorithm is repeatedly updated so that the required velocity error decreases.

Although physically intuitive and computationally inexpensive, such a law may also be interpreted as an approximation to an optimal steering law.

Linear-Tangent Steering¶

In several simplified ascent and intercept problems, optimality conditions yield a steering law for which the tangent of the steering angle is linear in time:

This is linear-tangent steering.

The significance is that a simple explicit law can emerge from optimal control theory and can then be implemented in real time without solving the full optimal control problem online.

Spacecraft Reorientation¶

Spacecraft attitude reorientation may require much more precise guidance than launch ascent.

Typical objectives include:

• minimum maneuver time,

• minimum fuel,

• minimum reaction-wheel momentum buildup,

• minimum structural excitation,

• terminal pointing accuracy.

The optimal control structure may involve:

• bang-bang control,

• singular arcs,

• actuator saturation,

• nonlinear rotational dynamics,

• state constraints.

Because pointing accuracy is critical, approximations must be designed and verified carefully.

Minimum-Time and Minimum-Fuel Guidance¶

For minimum time,

For minimum fuel,

or a related propellant-consumption model.

These objectives lead to different reference controls.

Thus, “best” has no meaning unless the performance criterion is specified.

What Does “Better” Mean?¶

Whenever one trajectory is judged better than another, an implicit performance measure is being used.

A formal optimization problem makes that measure explicit:

Even when engineers judge solutions qualitatively, they are implicitly considering criteria such as:

• time,

• energy,

• fuel,

• safety,

• smoothness,

• comfort,

• robustness,

• tracking accuracy,

• computational effort.

Role of Necessary Conditions¶

The calculus of variations and Pontryagin’s minimum principle produce conditions that an optimal solution must satisfy.

These conditions do not generally provide a simple algorithm for obtaining the solution.

They are:

• necessary rather than automatically sufficient;

• often analytically solvable only for special problems;

• frequently converted into difficult two-point boundary-value problems.

Their value is theoretical insight, structural characterization, and verification.

Role of Numerical Methods¶

Direct and indirect numerical methods generate approximations to the optimal reference trajectory.

Examples include:

• single shooting,

• multiple shooting,

• direct transcription,

• collocation,

• pseudospectral methods,

• sequential quadratic programming,

• interior-point methods.

The objective is ultimately to produce

that can be passed to the feedback controller.

Why Numerical Optimal Control May Be Too Slow¶

A high-fidelity optimal control problem may contain:

• nonlinear dynamics,

• hundreds or thousands of decision variables,

• path constraints,

• phase changes,

• discrete events,

• uncertainty,

• expensive simulation models.

A general nonlinear programming solver may require seconds, minutes, hours, or longer.

If the physical system evolves faster than the optimizer, the method cannot be used directly for online guidance.

Strategies for Real-Time Implementation¶

Several approaches can make optimal guidance practical.

Offline Optimization¶

Solve the problem before operation and store the resulting trajectory.

Trajectory Libraries¶

Precompute many trajectories and select or interpolate among them online.

Reduced-Order Models¶

Use simplified dynamics during online optimization.

Explicit Guidance Laws¶

Fit a low-order analytical law to high-fidelity optimal solutions.

Warm Starting¶

Initialize the current optimization using the previous solution.

Model Predictive Control¶

Solve a shorter-horizon problem repeatedly.

Surrogate Modeling¶

Replace expensive simulation models with fast approximations.

Policy Approximation¶

Approximate the optimal feedback law using regression, neural networks, or basis functions.

Navigation Errors and Estimation Errors¶

Measurements are always imperfect. Common error sources include:

• sensor noise,

• bias,

• scale-factor errors,

• misalignment,

• drift,

• delays,

• environmental disturbances,

• model mismatch.

State estimation filters these measurements and combines them with the model and control history.

If navigation errors become large, guidance may generate the wrong reference because it is solving the problem from an incorrect initial state estimate.

Interaction among Guidance, Estimation, and Control¶

The three subsystems are coupled:

• guidance depends on the estimated state;

• control depends on the reference and estimated state;

• estimation depends on the measured output and applied control;

• the plant responds to the total control;

• new measurements close the loop.

Poor performance in one subsystem can degrade the entire architecture.

Hierarchical View of the System¶

The system may be viewed hierarchically:

1. Mission level: select objectives and terminal goals.

2. Guidance level: generate feasible reference trajectories.

3. Control level: track the references.

4. Actuation level: produce physical forces and moments.

5. Navigation and estimation level: determine the current state.

Optimal control primarily operates at the guidance level, although optimal design methods can also be used at the control and estimation levels.

Open-Loop and Closed-Loop Terminology¶

Care is required when using the term open loop.

An optimal trajectory may be open loop at the guidance level while the vehicle remains closed loop at the control level.

For example:

• the reference trajectory is computed once;

• the feedback controller tracks it continuously;

• the guidance problem is never resolved.

The overall system contains feedback, but the optimal reference generation remains open loop.

Midcourse Correction¶

A midcourse correction occurs when the reference is recomputed after a deviation.

At correction time $t_c$:

1. navigation determines the current estimated state $\hat{\boldsymbol{x}}(t_c)$;

2. guidance solves a new problem from $\hat{\boldsymbol{x}}(t_c)$ to the target;

3. a new reference trajectory is generated;

4. the controller tracks the updated reference.

This is a practical implementation of Bellman’s principle of optimality.

Design Questions¶

When embedding optimal control into a complete system, ask:

1. What variables can be measured?

2. What states must be estimated?

3. How often is guidance updated?

4. How fast can the optimization be solved?

5. What model fidelity is required?

6. How large are the expected disturbances?

7. What constraints must always be respected?

8. Is exact optimality necessary?

9. What feedback controller will track the reference?

10. How will the full architecture be verified?

Verification of the Complete Architecture¶

Verification should include:

• reference feasibility,

• tracking error,

• state-estimation error,

• actuator saturation,

• robustness to disturbances,

• sensitivity to model mismatch,

• guidance update timing,

• computational latency,

• terminal-condition accuracy,

• safety-constraint satisfaction.

Summary¶

1. A complete control system includes plant dynamics, measurement, navigation, estimation, control, and guidance.

2. Navigation determines where the system is.

3. Guidance determines where the system should go.

4. Control generates corrective actuator commands.

5. Estimation reconstructs the state from measurements and control history.

6. Optimal control naturally generates reference state and control trajectories.

7. Therefore, optimal control is best interpreted as optimal guidance.

8. Guidance is an outer-loop function; tracking control is an inner-loop function.

9. An optimal reference may be generated once or repeatedly.

10. Repeated replanning produces receding-horizon or midcourse-correction behavior.

11. Exact optimality may be sacrificed for speed and implementability.

12. Practical guidance often uses simplified laws derived from or validated against optimal control theory.

13. Launch-vehicle ascent and spacecraft reorientation illustrate different levels of guidance complexity.

14. Numerical optimal control is valuable only when it can deliver a reference in time for the physical system.

Connection. Before invoking a large numerical transcription, analytical reduction can expose a low-dimensional shooting problem and improve both insight and robustness.