# Controls Glossary

bang-bang control

A very simple, no-tuning-required closed-loop control technique. It simply ”turns on“ the control effort when the process variable is too small, and ”turns off“ the control effort when the process variable is too big. It works well in some cases, but not all. See ”Bang-bang“ control on Wikipedia for more info.

Cartesian coordinate system

A set of points in space where each point is described by a set of numbers, indicating its coordinates within that space. These coordinates are an expression of the orthogonal distance of each point from a set of fixed, orthogonal axes (IE, a ”rectangular“ system). 2-dimension and 3-dimension spaces are most common in FRC (and likely what was learned in algebra 1), but any number of dimensions is theoretically possible. See Cartesian coordinate system on Wikipedia for more info.

churning losses

Complex friction-like forces arising from the fact that when gears and bearings rotate, they must displace liquid lubricant. This reduces the efficiency of rotating mechanisms.

control signal

The driving signal sent to a plant by a controller, usually quantified as a voltage.

control effort

Control signal

control law

A mathematical formula that generates inputs to drive a system to a desired state, given the current state. A common example is the control law $$\mathbf{u} = \mathbf{K(r - x)}$$

controller

Used in position or negative feedback with a plant to bring about a desired system state by driving the difference between a reference signal and the output to zero.

convolution

A mathematical operation that calculates a weighted moving average of one function, with the weights assigned by a second function. A common way to ”filter“ sensor input is to apply a convolution to it, using a carefully-chosen filtering function. See convolution. on Wikipedia for more info.

counter-electromotive force

A voltage generated in a spinning motor. The voltage is a result of the fact that has a coil of wire rotating near a magnet. See Counter-electromotive_force on Wikipedia for more info.

current

The flow of electrons through a conductor. Current is described with a unit of ”Amps“ (or simply ”A“), and is measured at a single point in a circuit. One amp is equal to $$6241509074000000000$$ electrons moving past the measurement point in one second.

dynamics

A branch of physics concerned with the motion of bodies under the action of forces. In modern control, systems evolve according to their dynamics.

derivative

A mathematical operation which evaluates the ”rate-of-change“ of a function at a given point. See derivative on Wikipedia for more info.

error

Reference minus an output or state.

An iterative process of finding a specific value within a wide search range by applying a multiplicative factor to the search value. See exponential search on Wikipedia for more info.

exponential smoothing

A very common way to implement a simple low-pass filter, using an exponential window function in a convolution with an input signal. The convolution operation simplifies down to a very simple set of math operations on the current input and previous output. See exponential smoothing on Wikipedia for more info.

gain

A scalar value that relates the magnitude of an input signal to the magnitude of an output signal. For example, gain in output = gain * input. A gain greater than one would amplify an input signal, while a gain less than one would dampen an input signal. A negative gain would negate the input signal.

Gaussian distribution

A special mathematical function that describes distributions of averages. The graph of a Gaussian function is a ”bell curve“ shape. This function is described by its mean (the location of the ”peak“ of the bell curve) and variance (a measure of how ”spread out“ the bell curve is). See Gaussian distribution on Wikipedia for more info.

The derivative, but applied to a function with multiple inputs. As a result, the output is both the magnitude of the rate of change, and the vector direction along which it occurs.

hidden state

A state that cannot be directly measured, but whose dynamics can be related to other states.

input

An input to the plant (hence the name) that can be used to change the plant’s state.

• Ex. A flywheel will have 1 input: the voltage of the motor driving it.

• Ex. A drivetrain might have 2 inputs: the voltages of the left and right motors.

Inputs are often represented by the variable $$\mathbf{u}$$, a column vector with one entry per input to the system.

least-squares regression

A curve-fitting technique which picks a curve to minimizes the square of the error between the fitted curve, and the actual measured data. See ordinary least-squares regression on Wikipedia for more info.

LQR

Linear-Quadratic Regulator - A feedback control scheme which seeks to operate a system in a ”most optimal“ or ”lowest cost“ manner, in the sense of minimizing the square of some ”cost function“ that represents a combination of system error and control effort. This requires an accurate mathematical model of the system being controlled, and function describing the ”cost“ of any given system state. See LQR on Wikipedia for more info.

measurement

Measurements are outputs that are measured from a plant, or physical system, using sensors.

model

A set of mathematical equations that reflects some aspect of a physical system’s behavior.

observer

In control theory, a system that provides an estimate of the internal state of a given real system from measurements of the input and output of the real system. WPILib includes a Kalman Filter class for observing linear systems, and ExtendedKalmanFilter and UnscentedKalmanFilter classes for nonlinear systems.

orthogonal

Having the property of being independent, or lacking mutual influence. For example, two lines are orthogonal if moving any number of units along one line causes zero displacement along the other line. In a cartesian coordinate system, orthogonal lines are often said to have 90-degree angles between each other.

output

Measurements from sensors. There can be more measurements then states. These outputs are used in the ”correct“ step of Kalman Filters.

• Ex. A flywheel might have 1 output from a encoder that measures it’s velocity.

• Ex. A drivetrain might use solvePNP and V-SLAM to find it’s x/y/heading position on the field. It’s fine that there are 6 measurements (solvePNP x/y/heading and V-SLAM x/y/heading) and 3 states (robot x/y/heading).

Outputs of a system are often represented using the variable $$\mathbf{y}$$, a column vector with one entry per output (or thing we can measure). For example, if our system had states for velocity and acceleration but our sensor could only measure velocity, our, our output vector would only include the system's velocity.

phase portrait

A graph of a function’s value and its derivative as they change in time, given some initial starting conditions. They are useful for analyzing system behavior (stable/unstable operating points, limit cycles, etc.) given a certain set of parameters or starting conditions. See phase portrait on Wikipedia for more info.

PID

Proportional-Integral-Derivative - A feedback controller which calculates a control signal from a weighted sum of the error, the rate of change of the error, and an accumulated sum of previous errors. See PID controller. on Wikipedia for more info.

plant

The system or collection of actuators being controlled.

process variable

The term used to describe the output of a plant in the context of PID control.

r-squared

A statistical measurement of how well a model predicts a set of data, representing the fraction of the observed variation in the independent variable that is accurately predicted by the model. The value typically runs from 0.0 (a terrible fit, equivalent to just guessing the average value of your independent variable) to 1.0 (a perfect fit). See Coefficient_of_determination on Wikipedia for more info.

reference

The desired state. This value is used as the reference point for a controller’s error calculation.

rise time

The time a system takes to initially reach the reference after applying a step input.

RMSE

Root Mean Squared Error - Statistical measurement of how well a curve is fit to a set of data. It is calculated as the square root of the average (mean) of the squares of all the errors between the actual sample and the curve fit. It has units of the original input data. See Root Mean Squared Error on Wikipedia for more info.

setpoint

The term used to describe the reference of a PID controller.

settling time

The time a system takes to settle at the reference after a step input is applied.

signum function

A non-continuous function that expresses the ”sign“ of its input. It is equal to -1 for all negative input numbers, 0 for an input of 0, and 1 for all positive input numbers. See signum function, on Wikipedia for more info.

state

A characteristic of a system (e.g., velocity) that can be used to determine the system’s future behavior. In state-space notation, the state of a system is written as a column vector describing it’s position in state-space.

• Ex. A drivetrain system might have the states $$\begin{bmatrix}x \\ y \\ \theta \end{bmatrix}$$ to describe it’s position on the field.

• Ex. An elevator system might have the states $$\begin{bmatrix} \text{position} \\ \text{velocity} \end{bmatrix}$$ to describe its current height and velocity.

A system’s state is often represented by the variable $$\mathbf{x}$$, a column vector with one entry per state.

statistically robust

The property of a data processing algorithm which makes it resilient to a noisy or outlier-prone data set. Designing statistically robust algorithms on robots is important because real-world sensor data can often be unpredictable, but unexpected robot behavior is never desirable. See Robust Statistics on Wikipedia for more info.

Error after system reaches equilibrium.

step input

A system input that is $$0$$ for $$t < 0$$ and a constant greater than $$0$$ for $$t \geq 0$$. A step input that is $$1$$ for $$t \geq 0$$ is called a unit step input.

step response

The response of a system to a step input.

system

A term encompassing a plant and it’s interaction with a controller and observer, which is treated as a single entity. Mathematically speaking, a system maps inputs to outputs through a linear combination of states.

system identification

The process of capturing a systems dynamics in a mathematical model using measured data. The SysId toolsuite uses system identification to find kS, kV and kA terms.

system response

The behavior of a system over time for a given input.

voltage

The measurement of how much an electric field is ”pushing“ electrons through a circuit. It is sometimes called ”Electromotive Force“, or ”EMF“. It is measured in units of ”Volts“. It always is defined between two points in a circuit. If one electron travels between two points that have one volt of EMF between them, it will have been accelerated to the point of having $$\frac{1}{6241509074000000000}$$ joules of energy.

viscous drag

The force generated from an object moving relatively slowly through non-turbulent fluid. In this region, the force is roughly proportional to the velocity of the object. It describes the most common type of ”air resistance“ an FRC robot would encounter, as well as losses in a gearbox from displacing grease. See Drag (physics) on Wikipedia for more info.

x-dot

$$\dot{\mathbf{x}}$$, or x-dot: the derivative of the state vector $$\mathbf{x}$$. If the system had just a velocity state, then $$\dot{\mathbf{x}}$$ would represent the system's acceleration.

x-hat

$$\hat{\mathbf{x}}$$, or x-hat: the estimated state of a system, as estimated by an observer.