The Central Idea

A control system can be organized as

\[ \boxed{ \text{system state} +\text{measurements} +\text{controller} +\text{actuation} +\text{feedback} \longrightarrow \text{controlled behavior} }. \]

The model predicts how the system evolves, the controller chooses inputs, the actuators apply those inputs, and sensors report what actually happened.

Open-Loop and Closed-Loop Control

An open-loop controller applies an input without using measured output to correct the action.

A closed-loop controller uses feedback:

\[ \boxed{ \text{reference} \rightarrow \text{controller} \rightarrow \text{system} \rightarrow \text{measurement} \rightarrow \text{controller} }. \]

Feedback allows the controller to respond to disturbances, uncertainty, and modeling error.

State Variables

A state vector contains the minimum information needed, together with future inputs, to determine future evolution under the model:

\[ \boxed{ \mathbf{x}(t) = \begin{bmatrix} x_1(t)\\ x_2(t)\\ \vdots\\ x_n(t) \end{bmatrix} }. \]

Examples include position, velocity, temperature, motor current, battery charge, joint angle, or angular velocity.

Inputs and Outputs

The control input is commonly written \(\mathbf{u}(t)\), while measured or controlled outputs are written \(\mathbf{y}(t)\).

A general nonlinear model is

\[ \boxed{ \dot{\mathbf{x}} = \mathbf{f}(\mathbf{x},\mathbf{u},t) } \] \[ \boxed{ \mathbf{y} = \mathbf{h}(\mathbf{x},\mathbf{u},t) }. \]

Linear State-Space Models

A continuous-time linear time-invariant model has the form

\[ \boxed{ \dot{\mathbf{x}} = A\mathbf{x} + B\mathbf{u} } \] \[ \boxed{ \mathbf{y} = C\mathbf{x} + D\mathbf{u} }. \]

The matrices \(A,B,C,D\) encode system dynamics, input effects, measured outputs, and direct feedthrough.

Discrete-Time State-Space Models

Digital controllers often use discrete-time models:

\[ \boxed{ \mathbf{x}_{k+1} = A_d\mathbf{x}_k + B_d\mathbf{u}_k } \] \[ \boxed{ \mathbf{y}_k = C_d\mathbf{x}_k + D_d\mathbf{u}_k }. \]

The sampling interval becomes part of the model.

Equilibria

An equilibrium state \(\mathbf{x}^*\) with constant input \(\mathbf{u}^*\) satisfies

\[ \boxed{ \mathbf{f}(\mathbf{x}^*,\mathbf{u}^*)=0 }. \]

Many controllers are designed to stabilize a desired equilibrium or track a time-varying reference around it.

Linearization

A nonlinear system can often be approximated near an operating point by a linear model:

\[ \boxed{ \delta\dot{\mathbf{x}} = A\delta\mathbf{x} + B\delta\mathbf{u} }. \]

The matrices are Jacobians evaluated at the operating point:

\[ A= \left. \frac{\partial\mathbf{f}}{\partial\mathbf{x}} \right|_*, \qquad B= \left. \frac{\partial\mathbf{f}}{\partial\mathbf{u}} \right|_*. \]

Stability

Stability asks whether small deviations from a desired state shrink, remain bounded, or grow.

For the linear system

\[ \dot{\mathbf{x}}=A\mathbf{x}, \]

asymptotic stability requires all eigenvalues of \(A\) to have negative real parts.

Discrete-Time Stability

For

\[ \mathbf{x}_{k+1}=A_d\mathbf{x}_k, \]

asymptotic stability requires every eigenvalue to lie strictly inside the unit circle:

\[ \boxed{ |\lambda_i|<1 }. \]

Feedback Control

A simple state-feedback law is

\[ \boxed{ \mathbf{u} = -K\mathbf{x} }. \]

Substituting into the state equation gives

\[ \dot{\mathbf{x}} = (A-BK)\mathbf{x}. \]

The controller changes the closed-loop dynamics through the matrix \(A-BK\).

Reference Tracking

A controller often seeks to make output \(y(t)\) follow a desired reference \(r(t)\). The tracking error is

\[ \boxed{ e(t)=r(t)-y(t) }. \]

Controller design determines how quickly and accurately this error is reduced.

Proportional Control

A proportional controller uses

\[ \boxed{ u(t)=K_Pe(t) }. \]

Larger \(K_P\) often produces stronger corrective action, but excessive gain can produce oscillation, overshoot, or instability.

Integral Control

Integral action accumulates error:

\[ \boxed{ u_I(t) = K_I \int_0^t e(\tau)\,d\tau }. \]

Integral control can eliminate steady-state error in many systems but may also increase overshoot or create windup when actuators saturate.

Derivative Control

Derivative action responds to the rate of change of error:

\[ \boxed{ u_D(t) = K_D\frac{de}{dt} }. \]

It can add damping but is sensitive to measurement noise and therefore often requires filtering.

PID Control

A PID controller combines proportional, integral, and derivative terms:

\[ \boxed{ u(t) = K_Pe(t) + K_I\int_0^t e(\tau)\,d\tau + K_D\frac{de}{dt} }. \]

PID control is widely used because it can achieve useful performance without a highly detailed model.

Actuator Saturation

Real actuators have limits:

\[ \boxed{ u_{\min} \le u(t) \le u_{\max} }. \]

Ignoring these limits can make simulated performance unattainable in the real system.

Integral Windup

When an actuator saturates, the integral term may continue accumulating error even though additional control effort cannot be applied.

Anti-windup strategies limit or reset the integral state when saturation occurs.

Transfer Functions

For linear time-invariant systems with zero initial conditions, the transfer function relates Laplace-domain input and output:

\[ \boxed{ G(s)=\frac{Y(s)}{U(s)} }. \]

Transfer functions emphasize input-output behavior rather than internal state.

First-Order Systems

A common first-order transfer function is

\[ \boxed{ G(s) = \frac{K}{\tau s+1} }. \]

The gain \(K\) controls steady-state scale, while \(\tau\) sets the response time.

Step Response of a First-Order System

For a unit step input,

\[ \boxed{ y(t) = K\left(1-e^{-t/\tau}\right) }. \]

At \(t=\tau\), the response reaches about \(63.2\%\) of its final value.

Second-Order Systems

A standard second-order transfer function is

\[ \boxed{ G(s) = \frac{\omega_n^2} {s^2+2\zeta\omega_ns+\omega_n^2} }. \]

The natural frequency \(\omega_n\) sets the time scale, while damping ratio \(\zeta\) strongly influences overshoot and oscillation.

Damping Regimes

Controller design often seeks a compromise between fast response and excessive oscillation or overshoot.

Poles and Zeros

Poles are values of \(s\) that make the transfer-function denominator zero. Zeros make the numerator zero.

Pole locations strongly determine stability and transient behavior.

Frequency Response

Substituting

\[ s=j\omega \]

gives the steady sinusoidal frequency response

\[ \boxed{ G(j\omega) }. \]

Magnitude and phase describe how different frequencies are amplified and shifted.

Bandwidth

Bandwidth provides a rough measure of how rapidly a closed-loop system can respond to changing commands.

Higher bandwidth can improve responsiveness but may amplify noise, excite neglected dynamics, or require more actuator effort.

Disturbance Rejection

Controllers are often designed not only to track commands but also to reject disturbances.

Examples include wind on a drone, road disturbances on a vehicle, load changes on a motor, or external force on a robotic arm.

Robustness

A robust controller maintains acceptable performance despite modeling error, parameter variation, disturbances, and uncertainty.

High nominal performance with poor robustness can produce a fragile system.

Controllability

Controllability asks whether available inputs can move the state throughout the relevant state space.

For a linear system, the controllability matrix is

\[ \boxed{ \mathcal{C} = \begin{bmatrix} B&AB&A^2B&\cdots&A^{n-1}B \end{bmatrix} }. \]

Full rank implies controllability.

Observability

Observability asks whether the internal state can be reconstructed from measured outputs.

For a linear system,

\[ \boxed{ \mathcal{O} = \begin{bmatrix} C\\ CA\\ CA^2\\ \vdots\\ CA^{n-1} \end{bmatrix} }. \]

Full rank implies observability.

State Estimation

Not every state variable can be measured directly. An estimator combines model predictions with sensor measurements to estimate hidden state:

\[ \boxed{ \text{model prediction} + \text{measurement correction} \longrightarrow \text{state estimate} }. \]

Luenberger Observer

A linear observer can be written

\[ \boxed{ \dot{\hat{\mathbf{x}}} = A\hat{\mathbf{x}} + B\mathbf{u} + L(\mathbf{y}-C\hat{\mathbf{x}}) }. \]

The term \(\mathbf{y}-C\hat{\mathbf{x}}\) is the measurement residual used to correct the estimate.

Kalman Filtering

The Kalman filter is a probabilistic state estimator for linear systems with Gaussian noise assumptions.

A discrete model may be written

\[ \mathbf{x}_{k+1} = A\mathbf{x}_k + B\mathbf{u}_k + \mathbf{w}_k, \] \[ \mathbf{y}_k = C\mathbf{x}_k + \mathbf{v}_k, \]

where \(\mathbf{w}_k\) and \(\mathbf{v}_k\) represent process and measurement noise.

Prediction and Correction

Kalman filtering alternates between

\[ \boxed{ \text{predict} \rightarrow \text{measure} \rightarrow \text{correct} }. \]

The relative trust placed in model and measurement depends on their estimated uncertainty.

Sensors

Robots and controllers may use encoders, inertial sensors, cameras, lidar, radar, force sensors, pressure sensors, temperature sensors, GPS, or other instruments.

Sensor bandwidth, bias, noise, drift, latency, saturation, and calibration all affect the control model.

Actuators

Actuators convert control commands into physical action. Examples include electric motors, hydraulic cylinders, pneumatic systems, thrusters, brakes, heaters, valves, and steering mechanisms.

Actuator dynamics and limits should be included when they materially affect performance.

Robot Configuration

A robot configuration describes its geometric state. For an \(n\)-joint robot, generalized coordinates may be written

\[ \boxed{ \mathbf{q} = \begin{bmatrix} q_1&q_2&\cdots&q_n \end{bmatrix}^{T} }. \]

Revolute joints use angles; prismatic joints use linear displacement.

Forward Kinematics

Forward kinematics maps joint coordinates to an end-effector pose:

\[ \boxed{ \mathbf{x} = f(\mathbf{q}) }. \]

The function depends on robot geometry and joint arrangement.

Inverse Kinematics

Inverse kinematics asks for joint coordinates that produce a desired pose:

\[ \boxed{ f(\mathbf{q}) = \mathbf{x}_{d} }. \]

Solutions may be multiple, nonexistent, or constrained by joint limits and collisions.

The Jacobian

The robot Jacobian maps joint velocity to end-effector velocity:

\[ \boxed{ \dot{\mathbf{x}} = J(\mathbf{q}) \dot{\mathbf{q}} }. \]

It also plays a central role in force relationships, singularities, and differential inverse kinematics.

Singularities

A kinematic singularity occurs when the Jacobian loses rank.

Near a singularity, some end-effector motions may require very large joint velocities, and control can become numerically or physically difficult.

Robot Dynamics

A standard manipulator dynamics model is

\[ \boxed{ M(\mathbf{q})\ddot{\mathbf{q}} + C(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}} + \mathbf{g}(\mathbf{q}) = \boldsymbol{\tau} }. \]

Here \(M\) is the inertia matrix, \(C\) represents velocity-dependent effects, \(\mathbf{g}\) gravity, and \(\boldsymbol{\tau}\) applied joint torques.

Friction and Unmodeled Dynamics

Real robots also experience joint friction, compliance, backlash, actuator delay, cable forces, payload variation, and structural flexibility.

These effects may need explicit models or robust control margins.

Trajectory Tracking

A robot often follows a desired trajectory

\[ \mathbf{q}_d(t), \qquad \dot{\mathbf{q}}_d(t), \qquad \ddot{\mathbf{q}}_d(t). \]

Tracking error is

\[ \boxed{ \mathbf{e} = \mathbf{q}_d-\mathbf{q} }. \]

Computed-Torque Control

If the robot dynamics are modeled accurately, a controller can compensate for modeled nonlinear dynamics:

\[ \boxed{ \boldsymbol{\tau} = M(\mathbf{q})\mathbf{v} + C(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}} + \mathbf{g}(\mathbf{q}) }. \]

The auxiliary input \(\mathbf{v}\) can then be chosen to impose desired error dynamics.

Motion Planning

Motion planning searches for a path from a start configuration to a goal while respecting obstacles and constraints:

\[ \boxed{ \mathbf{q}(0)=\mathbf{q}_{0}, \qquad \mathbf{q}(T)=\mathbf{q}_{g} }. \]

Planning occurs in configuration space rather than only in ordinary physical space.

Configuration Space

Each point in configuration space corresponds to a complete robot configuration.

Obstacles in physical space induce forbidden regions in configuration space, often with complicated geometry.

Path Planning versus Trajectory Planning

A path specifies geometry. A trajectory specifies geometry together with time.

Trajectory planning must satisfy velocity, acceleration, torque, power, and sometimes jerk limits.

Polynomial Trajectories

Smooth point-to-point motion can be generated with polynomial functions. A cubic trajectory may be written

\[ \boxed{ q(t) = a_0+a_1t+a_2t^2+a_3t^3 }. \]

Coefficients are selected from boundary conditions on position and velocity.

Mobile-Robot Kinematics

A simple planar unicycle model is

\[ \boxed{ \dot{x}=v\cos\theta } \] \[ \boxed{ \dot{y}=v\sin\theta } \] \[ \boxed{ \dot{\theta}=\omega }. \]

The controls are forward speed \(v\) and angular speed \(\omega\).

Nonholonomic Constraints

Wheeled robots often cannot move freely in every instantaneous direction. A standard no-side-slip constraint is

\[ \boxed{ -\sin\theta\,\dot{x} + \cos\theta\,\dot{y} =0 }. \]

Such constraints make motion planning different from ordinary point motion.

Localization

Localization estimates a robot's pose from motion models and sensor data.

A probabilistic formulation seeks

\[ \boxed{ p(\mathbf{x}_t \mid \mathbf{z}_{1:t}, \mathbf{u}_{1:t}) }. \]

Here \(\mathbf{z}\) denotes measurements and \(\mathbf{u}\) control inputs.

Dead Reckoning

Integrating wheel motion or inertial measurements gives an estimate of position and orientation.

Small bias and noise accumulate over time, producing drift unless external measurements provide correction.

Sensor Fusion

Sensor fusion combines measurements with complementary strengths.

For example, inertial sensors provide rapid local motion information, while GPS or vision can provide slower absolute position corrections.

Simultaneous Localization and Mapping

SLAM estimates both robot pose and an unknown map from sensor observations.

The robot needs a map to localize, while building the map requires estimates of where the robot was when observations were made.

Optimal Control

Optimal control selects an input trajectory to minimize an objective such as

\[ \boxed{ J = \int_0^T L(\mathbf{x},\mathbf{u},t)\,dt + \Phi(\mathbf{x}(T)) }. \]

The objective may penalize tracking error, energy, time, control effort, risk, or terminal error.

Linear Quadratic Regulator

For a linear system, the LQR objective is

\[ \boxed{ J = \int_0^\infty \left( \mathbf{x}^{T}Q\mathbf{x} + \mathbf{u}^{T}R\mathbf{u} \right)dt }. \]

The matrices \(Q\) and \(R\) define the tradeoff between state deviation and control effort.

Model Predictive Control

Model predictive control repeatedly solves a finite-horizon optimization problem using the current estimated state.

At each step,

\[ \boxed{ \text{estimate state} \rightarrow \text{optimize future controls} \rightarrow \text{apply first control} \rightarrow \text{repeat} }. \]

MPC is valuable because constraints can be included explicitly.

Control Constraints

Model predictive control can enforce constraints such as

\[ \boxed{ \mathbf{x}_{\min} \le \mathbf{x}_k \le \mathbf{x}_{\max} } \]

and

\[ \boxed{ \mathbf{u}_{\min} \le \mathbf{u}_k \le \mathbf{u}_{\max} }. \]

Control and Planning

Planning selects a feasible or optimal future route or trajectory. Control responds continuously to deviations between the plan and the actual system.

A practical autonomous system often requires both.

Hybrid Systems

Some systems combine continuous dynamics with discrete mode changes.

Examples include gear changes, contact events, walking phases, robotic grasping, safety interlocks, and switching converters.

These are modeled as hybrid dynamical systems.

Contact and Collision

Robotic interaction with the environment introduces unilateral constraints, friction, impacts, and discontinuous forces.

Contact-rich manipulation often requires specialized numerical and control methods.

Force Control

Some robotic tasks regulate force rather than position.

A desired contact force \(F_d\) can define error

\[ \boxed{ e_F=F_d-F }. \]

Force control is important for assembly, polishing, gripping, and interaction with people or compliant environments.

Impedance Control

Impedance control shapes the dynamic relationship between motion and applied force. A desired one-dimensional behavior may be

\[ \boxed{ M_d\ddot e + B_d\dot e + K_de = F_{\text{ext}} }. \]

This allows a robot to behave compliantly rather than rigidly.

Sampling and Digital Control

Digital controllers measure and update at discrete times. If the sample period is \(T_s\), then the sample rate is

\[ \boxed{ f_s=\frac{1}{T_s} }. \]

Sampling must be fast enough to capture the dynamics relevant to control.

Delay

Computation, sensing, communication, and actuation introduce delay. A delayed input may appear as

\[ \boxed{ u(t-\tau) }. \]

Delay adds phase lag and can reduce stability margins.

Networked Control

Distributed robots, industrial systems, and autonomous fleets may communicate over networks.

Packet loss, delay, bandwidth limits, synchronization, and communication failures then become part of the control model.

Multi-Robot Systems

Several robots can coordinate through local rules, shared plans, or centralized optimization.

Models may address formation control, task allocation, collision avoidance, consensus, and cooperative transport.

Consensus

In a simple consensus model,

\[ \boxed{ \dot{x}_i = \sum_{j\in\mathcal{N}_i} a_{ij}(x_j-x_i) }. \]

Each agent adjusts toward neighboring agents. Network connectivity determines whether agreement can be reached.

System Identification

When dynamics are not known accurately, model parameters can be estimated from input-output data.

A general identification problem seeks

\[ \boxed{ \hat{\theta} = \arg\min_\theta \sum_k \|y_k-\hat y_k(\theta)\|^2 }. \]

Persistent Excitation

Parameters cannot be identified well if the experiment does not sufficiently excite the system dynamics.

Experimental input design is therefore part of system identification.

Adaptive Control

Adaptive control updates model or controller parameters during operation.

It is useful when system dynamics vary, but adaptation must be designed so that parameter updates do not destabilize the system.

Learning-Based Control

Data-driven models can estimate dynamics, disturbances, or control policies.

Learning can complement control theory, but safety and stability constraints are especially important when learned policies act directly on physical systems.

Reinforcement Learning

Reinforcement learning represents sequential decision-making through states, actions, rewards, and transitions.

A discounted return may be

\[ \boxed{ G_t = \sum_{k=0}^{\infty} \gamma^k r_{t+k} }. \]

The policy seeks actions that maximize expected cumulative reward.

Reward Design

A reward function encodes what the learning system is encouraged to do.

Poor reward design can produce behavior that maximizes the numerical objective while violating the actual engineering intent.

Simulation

Control and robotics development relies heavily on simulation because physical testing can be expensive, slow, or dangerous.

Simulation can test nominal operation, disturbances, sensor noise, actuator faults, collision cases, and extreme conditions.

Simulation-to-Reality Gap

A controller that performs well in simulation may fail in hardware because the model omitted friction, delay, compliance, sensor artifacts, unmodeled contact, or environmental variation.

Hardware validation remains essential.

Verification

Verification asks whether the equations, software, optimization, and numerical implementation are correct.

Useful checks include

Validation

Validation asks whether the model and controller adequately represent the real system for their intended use.

Relevant comparisons may include step response, tracking error, settling time, overshoot, energy use, disturbance rejection, path accuracy, estimator error, or behavior under payload and environmental variation.

Sensitivity Analysis

Important sensitivities may include performance with respect to

Uncertainty

Real control systems contain uncertainty in model parameters, disturbances, sensors, actuators, environment, contact, payload, and timing.

Robustness and safety analysis should reflect those uncertainties rather than only nominal dynamics.

Example: First-Order Response

A system has

\[ G(s)=\frac{2}{3s+1}. \]

Therefore,

\[ K=2, \qquad \tau=3. \]

For a unit step,

\[ y(t) = 2\left(1-e^{-t/3}\right). \]

The final value is \(2\), and after \(3\) time units the response has reached about \(63.2\%\) of that change.

Example: Proportional Control

Suppose the reference is

\[ r=10 \]

and the measured output is

\[ y=7. \]

Then

\[ e=10-7=3. \]

If \(K_P=4\),

\[ u=K_Pe=(4)(3) \] \[ \boxed{u=12}. \]

Example: Discrete Stability

Consider the scalar system

\[ x_{k+1}=0.8x_k. \]

Since

\[ |0.8|<1, \]

repeated updates drive \(x_k\) toward zero:

\[ \boxed{\text{the system is asymptotically stable}}. \]

Example: Mobile-Robot Motion

A robot has

\[ v=2\ \text{m/s}, \qquad \theta=0. \]

Then

\[ \dot{x}=2\cos0=2, \] \[ \dot{y}=2\sin0=0. \]

The robot moves along the positive \(x\)-direction at

\[ \boxed{2\ \text{m/s}}. \]

Example: Sampling Rate

If a digital controller samples every

\[ T_s=0.01\ \text{s}, \]

then

\[ f_s=\frac{1}{0.01} \] \[ \boxed{f_s=100\ \text{Hz}}. \]

Common Error: Designing Only for the Nominal Model

A controller tuned for one exact mass, friction value, or payload may perform poorly when the physical system differs.

Robustness should be tested across credible parameter variation.

Common Error: Ignoring Actuator Limits

A controller may demand torque, voltage, speed, or force that the actuator cannot supply.

Saturation can change the closed-loop behavior substantially.

Common Error: Ignoring Sensor Delay and Noise

Feedback depends on measurements. Delay, filtering, bias, and noise can reduce stability margins and degrade tracking.

Common Error: Confusing Stability with Good Performance

A stable system may still be too slow, too oscillatory, too inaccurate, or too energy intensive.

Stability is necessary for many applications but is not the only performance criterion.

Common Error: Ignoring Sampling

A controller designed in continuous time can behave differently after digital implementation if the sampling period is too large or delay is significant.

Common Error: Treating Kinematics as Dynamics

Kinematics describes geometric motion without accounting for forces or inertia. Dynamics determines how forces and torques produce acceleration.

A path that is kinematically feasible may still be dynamically infeasible.

Common Error: Ignoring Singularities

Near a robot singularity, inverse kinematic or velocity calculations can demand enormous joint motion.

Planning and control should detect and manage these configurations.

Common Error: Treating Simulation as Validation

Simulation verifies behavior within a model. It does not establish that the model adequately represents hardware.

Experimental comparison is required for physical validation.

A Reliable Modeling Procedure

  1. Define the control or robotic task and performance requirements.
  2. Choose state variables, inputs, outputs, disturbances, and constraints.
  3. Develop kinematic and dynamic models at the level needed for the task.
  4. Identify equilibria, operating regions, and relevant time scales.
  5. Assess stability, controllability, and observability.
  6. Select sensors, estimators, actuators, and controller structure.
  7. Include sampling, delay, saturation, noise, and uncertainty where material.
  8. Verify equations, numerical implementation, and optimization logic.
  9. Validate against physical measurements and hardware behavior.
  10. Perform robustness, sensitivity, failure, and safety testing before deployment.

Practice 1: Tracking Error

A reference is \(r=15\), and measured output is \(y=11\). Find the tracking error \(e=r-y\).

Practice 2: Proportional Control

If \(K_P=5\) and \(e=2.4\), find the proportional control input.

Practice 3: First-Order Response

A system has gain \(K=3\) and time constant \(\tau=2\). For a unit step, find the output at \(t=2\) using

\[ y(t)=K(1-e^{-t/\tau}). \]

Practice 4: Continuous-Time Stability

A two-state linear system has eigenvalues

\[ -2,\qquad-0.5. \]

Is the equilibrium asymptotically stable?

Practice 5: Discrete-Time Stability

A scalar system satisfies

\[ x_{k+1}=1.05x_k. \]

Is the origin asymptotically stable?

Practice 6: Sampling Rate

A controller samples every \(0.005\) second. Find the sampling frequency.

Practice 7: Mobile-Robot Velocity

A unicycle robot has

\[ v=3\ \text{m/s}, \qquad \theta=\frac{\pi}{2}. \]

Find \(\dot{x}\) and \(\dot{y}\).

Practice 8: Actuator Saturation

A controller requests \(u=18\), but the actuator limit is

\[ -10\le u\le10. \]

What input can the physical actuator apply?

Practice 9: LQR Tradeoff

In

\[ J = \int \left( \mathbf{x}^TQ\mathbf{x} + \mathbf{u}^TR\mathbf{u} \right)dt, \]

explain qualitatively what happens when the control-effort penalty \(R\) is increased substantially.

Practice 10: Choose the Better Model

A robotic arm tracks well in simulation but oscillates when carrying a heavy payload. The simulation assumed a fixed nominal mass and ideal actuators. Identify the modeling deficiencies that should be investigated.

Practice Answers

Practice 1

\[ e=r-y =15-11 \] \[ \boxed{e=4}. \]

Practice 2

\[ u=K_Pe =(5)(2.4) \] \[ \boxed{u=12}. \]

Practice 3

\[ y(2) = 3\left(1-e^{-2/2}\right) \] \[ = 3(1-e^{-1}) \] \[ \boxed{y(2)\approx1.90}. \]

Practice 4

Both eigenvalues have negative real parts:

\[ -2<0, \qquad -0.5<0. \]

Therefore,

\[ \boxed{\text{yes, the equilibrium is asymptotically stable}}. \]

Practice 5

The discrete-time eigenvalue is \(1.05\), and

\[ |1.05|>1. \]

Therefore,

\[ \boxed{\text{the origin is unstable}}. \]

Practice 6

\[ f_s=\frac{1}{0.005} \] \[ \boxed{f_s=200\ \text{Hz}}. \]

Practice 7

\[ \dot{x} = 3\cos\left(\frac{\pi}{2}\right) =0, \] \[ \dot{y} = 3\sin\left(\frac{\pi}{2}\right) =3. \] \[ \boxed{ \dot{x}=0, \qquad \dot{y}=3\ \text{m/s} }. \]

Practice 8

The request exceeds the actuator maximum, so the applied input saturates at

\[ \boxed{u=10}. \]

Practice 9

Increasing \(R\) makes control effort more expensive in the optimization. The controller generally becomes less aggressive, using smaller inputs and accepting slower or less exact state correction when necessary.

Practice 10

The model should include payload-dependent inertia, actuator torque and speed limits, and possibly friction, compliance, delay, and structural flexibility. The controller should then be tested for robustness across the expected payload range rather than only at the nominal model.

What to Remember

Where We Go Next

Control systems and robotics show how models support real-time estimation, decision-making, and feedback in dynamic physical systems. Many of those systems depend on extracting information from measured signals and images before control decisions can be made.

The next lesson, Imaging and Signal Processing, develops models of sampling, filtering, convolution, transforms, noise, reconstruction, frequency content, image formation, and computational inference from measured signals.