State-Feedback Controller

Discrete-time state-feedback controller with integral action

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Libraries:
Simscape / Electrical / Control / General Control

Description

The State-Feedback Controller block implements a discrete-time state-feedback controller with integral action. Use this block to control linear systems with single or multiple inputs and single or multiple outputs. The integral action serves to eliminate steady-state error in the controlled outputs. You can define the controller using a precomputed optimal gain or use the state-space model of your system to generate this gain using pole placement.

Equations

The integral of the tracking error, x_i, is an additional state that ensures zero steady-state error for the closed-loop system. The extended state vector is

$x_e=\left[\begin{array}{c}x\\ x_i\end{array}\right],$

Where:

• x is the state vector.

• x_i is the integral of the tracking error.

• x_e is the extended state vector.

Therefore, the control action is

$u=K{x}_e,$

Where:

• K is the feedback matrix, that is, the pole placement.

• u is the controller output.

Assumptions

System state measurement and estimation occur outside the controller.

Examples

Synchronous Machine State-Space Control

Control currents in a synchronous machine (SM) based traction drive using state-space control. A high-voltage battery feeds the SM through a controlled three-phase converter for the stator windings and through a controlled two-quadrant chopper for the rotor winding. An ideal angular velocity source provides the load. The SM operates below the base speed. At each sample instant, the torque request is converted to relevant current references using the zero d-axis control approach. A state-feedback controller controls the currents in the rotor reference frame. A Luenberger observer obtains the velocity-dependent feedforward pre-control terms. The simulation uses several torque steps in both motor and generator modes. The task scheduling is implemented as a Stateflow® state machine. The Scopes subsystem contains scopes that allow you to see the simulation results.

• Open Model

DC Motor Control (State-Feedback and Observer)

A state-feedback speed-control structure for a DC motor. A PWM controlled four-quadrant Chopper is used to feed the DC motor. The Control subsystem includes the state-feedback control loop, and the PWM generation. The state vector includes the rotor speed which is measured, and the dc motor current, which is estimated using an observer. Both the observer and state-feedback controller are synthesized by pole placement using the state-space model of the system. The total simulation time (t) is 4 seconds. At t = 1.5 seconds, the load torque increases. At t = 2.5 seconds, the reference speed is changed from 1000 rpm to 2000 rpm.

• Open Model

Ports

Input

r — Plant reference

scalar

Plant system reference signal.

Data Types: single | double

x — State vector

vector

Measured or estimated system state vector.

Data Types: single | double

Reset — Integrator reset

scalar

External reset signal (rising edge) for the integrator.

Data Types: Boolean

y — Plant output

scalar

Plant system output signal.

Data Types: single | double

Output

u — Controller output

scalar

Control system output signal.

Data Types: single | double

Parameters

State-feedback design — Controller generation

State-feedback gain (default) | Desired eigenvalues

Select the strategy for parameterizing controller gain:

• State-feedback gain — Specify the controller gain directly

• Desired eigenvalues — Specify the plant model and desired eigenvalues from which to generate the controller gain

State-feedback parameterization — State-feedback parameterization

Discrete-time (default) | Continuous-time

Select the strategy for parameterizing the state-space matrices and desired poles for the controller. The block implementation is discrete regardless of this parameterization.

Dependencies

To enable this parameter, set State-feedback design to Desired eigenvalues.

Controller matrix — Controller matrix

[1 1] (default) | matrix

Controller feedback matrix. To determine the controller matrix, if you have a license for Control System Toolbox™, use the lqr or lqi function.

Dependencies

To enable this parameter, set State-feedback design to State-feedback gain.

Discrete-time A matrix — A matrix in discrete time

1 (default) | real scalar or matrix

State matrix of the discrete-time state-space model. The A matrix must be square, with the number of rows and columns equal to the order of the system.

Dependencies

To enable this parameter, set State-feedback parameterization to Discrete-time.

Discrete-time B matrix — B matrix in discrete time

1 (default) | real scalar or matrix

Input matrix of the discrete-time state-space model. The B matrix must have the number of rows equal to the order of the system, and the number of columns equal to the number of system inputs.

Dependencies

To enable this parameter, set State-feedback parameterization to Discrete-time.

Discrete-time C matrix — C matrix in discrete time

1 (default) | real scalar or matrix

Output matrix of the discrete-time state-space model. The C matrix must have the number of rows equal the number of outputs of the system, and the number of columns equal to the order of the system.

Dependencies

To enable this parameter, set State-feedback parameterization to Discrete-time.

Discrete-time D matrix — D matrix in discrete time

1 (default) | real scalar or matrix

Feedthrough matrix of the discrete-time state-space model. The D matrix must have the number of rows equal to the number of system outputs, and the number of columns equal to the number of system inputs.

Dependencies

To enable this parameter, set State-feedback parameterization to Discrete-time.

Continuous-time A matrix — A matrix in continuous time

1 (default) | real scalar or matrix

State matrix of the continuous-time state-space model. The A matrix must be square, with the number of rows and columns equal to the order of the system.

Dependencies

To enable this parameter, set State-feedback parameterization to Continuous-time.

Continuous-time B matrix — B matrix in continuous time

1 (default) | real scalar or matrix

Input matrix of the continuous-time state-space model. The B matrix must have the number of rows equal to the order of the system, and the number of columns equal to the number of system inputs.

Dependencies

To enable this parameter, set State-feedback parameterization to Continuous-time.

Continuous-time C matrix — C matrix in continuous time

1 (default) | real scalar or matrix

Output matrix of the continuous-time state-space model. The C matrix must have the number of rows equal the number of outputs of the system, and the number of columns equal to the order of the system.

Dependencies

To enable this parameter, set State-feedback parameterization to Continuous-time.

Continuous-time D matrix — D matrix in continuous time

1 (default) | real scalar or matrix

Feedthrough matrix of the continuous-time state-space model. The D matrix must have the number of rows equal to the number of system outputs, and the number of columns equal to the number of system inputs.

Dependencies

To enable this parameter, set State-feedback parameterization to Continuous-time.

Discretization sample time — Discretization sample time

0.1 (default) | positive real number

Value used to discretize the state space matrices and also approximate the discrete-time eigenvalues.

Dependencies

To enable this parameter, set State-feedback parameterization to Continuous-time and Sample time (-1 for inherited) to -1.

Desired eigenvalues (discrete) — Observer eigenvalues

0 (default) | real vector

Specify the location of the eigenvalues to lie within the unit circle. The controller gain is then calculated based on these eigenvalues. The size of the vector must be equal to the system order plus the number of outputs.

Control action upper limit — u_max

5 (default) | scalar greater than the value of the Control action lower limit parameter

Upper limit for the control output signal.

Control action lower limit — u_min

0 (default) | scalar

Lower limit for the control output signal.

Sample time (-1 for inherited) — Sampling interval

-1 (default) | default value or a positive number

Time interval between samples. If the 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. For more information, see What Is Sample Time? and Specify Sample Time.

Extended Capabilities

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

Version History

Introduced in R2017b