Model Reference Adaptive Controller

Discrete-time PID-based model reference adaptive control

Library:
Simscape / Electrical / Control / General Control

Description

The Model Reference Adaptive Controller block implements discrete-time proportional-integral-derivative (PID) model reference adaptive control (MRAC). The three main components of an MRAC system are the reference model, the adjustment mechanism, and the controller.

Equations

The control equation is

$u_{p i d} (k) = [K_{p} + K_{i} \frac{T_{s} z}{z - 1} + K_{d} \frac{z - 1}{T_{s} z}] e (k),$

where:

u_pid is the controller output.
K_p is the proportional gain.
K_i is the integral gain.
K_d is the differential gain.
T_s is the sample time.
e is the error.

The reference model is the transfer function for the closed-loop system. This model captures the desired behavior of the closed-loop system. It is implemented as the discrete-time transfer function

$G_{m} (z) = \frac{B (z)}{A (z)} .$

The adaptation mechanism adjusts the control action based on the error between the plant output and the reference model output as

$θ = (y - y_{m}) y_{m} \frac{- γ T_{s} z}{z - 1},$

where:

θ is the adaptation parameter.
y is the plant output.
y_m is the reference model output.
γ is the learning rate.

Increasing the value of γ results in faster adaptation to plant changes.

The adjusted control signal, u, is

$u (k) = u_{p i d} (k) θ (k) .$

Ports

Input

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`r` — Plant reference, r
scalar

Plant system reference signal.

Data Types: single | double

`y` — Plant output, y
scalar

Plant system output signal.

Data Types: single | double

`Reset` — Integrator reset
scalar

External reset signal (rising edge) for the integrator.

Data Types: Boolean

Output

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`u` — Controller adjusted output, u
scalar

Adjusted control signal.

Data Types: single | double

Parameters

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Controller Parameters

`Proportional gain` — Controller proportional gain, K_p
`1` (default) | positive scalar

Proportional gain, K_p, of the controller.

`Integral gain` — Controller integral gain, K_i
`1` (default) | positive scalar

Integral gain, K_i, of the controller.

`Derivative gain` — Controller derivative gain, K_d
`0` (default)

Derivative gain, K_d, of the controller.

`Anti-windup gain` — Controller anti-windup gain
`1` (default) | positive scalar

Anti-windup gain of the controller.

`Use filtered derivative` — Filter option
`on` (default) | off

Choose whether to use a filter coefficient for the reference signal.

Dependencies

The Filter coefficient parameter is only visible when the Use filtered derivative check box is selected.

`Filter coefficient` — Reference signal filter coefficient
`100` (default)

Filter coefficient for the reference signal.

Dependencies

This parameter is only visible when the Use filtered derivative check box is selected.

`Control action upper limit` — Control signal upper limit, u_{pid_max}
`5` (default)

Upper bound for the control signal.

`Control action lower limit` — Control signal lower limit, u_{pid_min}
`-5` (default)

Lower bound for the control signal.

`Sample time (-1 for inherited)` — Block sample time
`-1` (default) | positive scalar

Time, in s, between consecutive block executions. During execution, the block produces outputs and, if appropriate, updates its internal state. For more information, see What Is Sample Time? and Specify Sample Time.

If this block is inside a triggered subsystem, inherit the sample time by setting this parameter to -1. If this block is in a continuous variable-step model, specify the sample time explicitly using a positive scalar.

Dependencies

If you set Sample time (-1 for inherited) to -1 and, in the Reference Model settings, set Model parameterization to Continuous-time, the Discretization sample time parameter becomes visible in the Reference Model settings.

Reference Model

`Model parameterization` — Discrete or continuous model option
`Discrete-time` (default) | `Continuous-time`

Mathematical model for the controller.

Dependencies

Choosing:

Discrete-time makes the discrete-time parameters visible.
Continuous-time makes the continuous-time parameters visible. Also, in the Controller Parameter settings, if Sample time (-1 for inherited) is set to -1, choosing this option makes the Discretization sample time parameter visible in the Reference Model settings.

`Discrete-time numerator` — Discrete-time transfer function numerator
`[0.01 -0.0099]` (default)

Numerator for the discrete-time transfer function.

Dependencies

Choosing Discrete-time for Model parameterization makes this parameter visible.

`Discrete-time denominator` — Discrete-time transfer function denominator
`[1 -1.9801 0.9802]` (default) | vector

Denominator for the discrete-time transfer function.

Dependencies

Choosing Discrete-time for Model parameterization makes this parameter visible.

`Discretization sample time` — Sample time for discretization
`0.01` (default) | `-1` or positive number

Time, in seconds, between consecutive discretizations. If block sample time is inherited, specify the discretization sample time explicitly.

Dependencies

This parameter is only visible when both of these conditions are met:

In the Control Parameters settings, Sample time (-1 for inherited) is set to -1.
In the Reference Model settings, Model parameterization is set to Continuous-time.

`Continuous-time numerator` — Continuous-time transfer function numerator
`[1 1]` (default) | vector

Numerator for the continuous-time transfer function.

Dependencies

Choosing Continuous-time for Model parameterization makes this parameter visible.

`Continuous-time denominator` — Continuous-time transfer function denominator
`[1 2 1]` (default) | vector

Denominator for the continuous-time transfer function.

Dependencies

Choosing Continuous-time for Model parameterization makes this parameter visible.

Adjustment Mechanism

`Learning rate` — γ
`0.5` (default)

Rate of adjustment to plant changes.

References

[1] Butler, H. Model-Reference Adaptive Control-From Theory to Practice. Upper Saddle River, NJ: Prentice Hall, 1992.

Documentation

Model Reference Adaptive Controller

Description

Equations

Ports

Input

r — Plant reference, r scalar

y — Plant output, y scalar

Reset — Integrator reset scalar

Output

u — Controller adjusted output, u scalar

Parameters

Controller Parameters

Proportional gain — Controller proportional gain, Kp 1 (default) | positive scalar

Integral gain — Controller integral gain, Ki 1 (default) | positive scalar

Derivative gain — Controller derivative gain, Kd 0 (default)

Anti-windup gain — Controller anti-windup gain 1 (default) | positive scalar

Use filtered derivative — Filter option on (default) | off

Dependencies

Filter coefficient — Reference signal filter coefficient 100 (default)

Dependencies

Control action upper limit — Control signal upper limit, upid_max 5 (default)

Control action lower limit — Control signal lower limit, upid_min -5 (default)

Sample time (-1 for inherited) — Block sample time -1 (default) | positive scalar

Dependencies

Reference Model

Model parameterization — Discrete or continuous model option Discrete-time (default) | Continuous-time

Dependencies

Discrete-time numerator — Discrete-time transfer function numerator [0.01 -0.0099] (default)

Dependencies

Discrete-time denominator — Discrete-time transfer function denominator [1 -1.9801 0.9802] (default) | vector

Dependencies

Discretization sample time — Sample time for discretization 0.01 (default) | -1 or positive number

Dependencies

Continuous-time numerator — Continuous-time transfer function numerator [1 1] (default) | vector

Dependencies

Continuous-time denominator — Continuous-time transfer function denominator [1 2 1] (default) | vector

Dependencies

Adjustment Mechanism

Learning rate — γ 0.5 (default)

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using Simulink® Coder™.

See Also

Simscape Electrical Documentation

Support

`r` — Plant reference, r
scalar

`y` — Plant output, y
scalar

`Reset` — Integrator reset
scalar

`u` — Controller adjusted output, u
scalar

`Proportional gain` — Controller proportional gain, K_p
`1` (default) | positive scalar

`Integral gain` — Controller integral gain, K_i
`1` (default) | positive scalar

`Derivative gain` — Controller derivative gain, K_d
`0` (default)

`Anti-windup gain` — Controller anti-windup gain
`1` (default) | positive scalar

`Use filtered derivative` — Filter option
`on` (default) | off

`Filter coefficient` — Reference signal filter coefficient
`100` (default)

`Control action upper limit` — Control signal upper limit, u_{pid_max}
`5` (default)

`Control action lower limit` — Control signal lower limit, u_{pid_min}
`-5` (default)

`Sample time (-1 for inherited)` — Block sample time
`-1` (default) | positive scalar

`Model parameterization` — Discrete or continuous model option
`Discrete-time` (default) | `Continuous-time`

`Discrete-time numerator` — Discrete-time transfer function numerator
`[0.01 -0.0099]` (default)

`Discrete-time denominator` — Discrete-time transfer function denominator
`[1 -1.9801 0.9802]` (default) | vector

`Discretization sample time` — Sample time for discretization
`0.01` (default) | `-1` or positive number

`Continuous-time numerator` — Continuous-time transfer function numerator
`[1 1]` (default) | vector

`Continuous-time denominator` — Continuous-time transfer function denominator
`[1 2 1]` (default) | vector

`Learning rate` — γ
`0.5` (default)

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.