Self-Conditioned [A,B,C,D]

Implement state-space controller in self-conditioned form

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Libraries:
Aerospace Blockset / GNC / Control

Description

The Self-Conditioned [A,B,C,D] block can be used to implement the state-space controller defined by

$$\left[\begin{array}{l}\dot{x}=Ax+Be\\ u=Cx+De\end{array}\right]$$

in the self-conditioned form

$$\begin{array}{l}\dot{z}=(A-HC)z+(B-HD)e+H{u}_{meas}\\ u_{dem}=Cz+De\end{array}$$

The input $$u_{meas}$$ is a vector of the achieved actuator positions, and the output $$u_{dem}$$ is the vector of controller actuator demands. In the case that the actuators are not limited, then $$u_{meas}=u_{dem}$$ and substituting the output equation into the state equation returns the nominal controller. In the case that they are not equal, the dynamics of the controller are set by the poles of A-HC.

Hence H must be chosen to make the poles sufficiently fast to track $$u_{meas}$$ but at the same time not so fast that noise on e is propagated to $$u_{dem}$$. The matrix H is designed by a callback to the Control System Toolbox™ command place to place the poles at defined locations.

Examples

Self-Conditioned Controller Comparison

Compare the implementation of a state-space controller [A,B,C,D] in a self-conditioned form versus a typical state-space controller [A,B,C,D].

• Open Script

Limitations

This block requires the Control System Toolbox license.

Ports

Input

e — Control error

vector

Control error, specified as a vector.

Data Types: double

u_meas — Achieved actuator positions

vector

Achieved actuator positions, specified as a vector.

Data Types: double

Output

u_dem — Actuator demands

vector

Actuator demands, specified as a vector.

Data Types: double

Parameters

A-matrix — A-matrix of the state-space implementation

[-1 -0.2;0 -3] (default) | array

A-matrix of the state-space implementation. The A-matrix should have three dimensions, the last one corresponding to the scheduling variable v. For example, if the A-matrix corresponding to the first entry of v is the identity matrix, then A(:,:,1) = [1 0;0 1];.

Programmatic Use

Block Parameter: Ak

Type: character vector

Values: vector

Default: '[-1 -0.2;0 -3]'

B-matrix — B-matrix of the state-space implementation

[1;1] (default) | array

B-matrix of the state-space implementation, specified as an array. The B-matrix should have three dimensions, the last one corresponding to the scheduling variable v. For example, if the B-matrix corresponding to the first entry of v is the identity matrix, then B(:,:,1) = [1 0;0 1];.

Programmatic Use

Block Parameter: Bk

Type: character vector

Values: vector

Default: '[1;1]'

C-matrix — C-matrix of the state-space implementation

[1 0] (default) | array

C-matrix of the state-space implementation, specified as an array. The C-matrix should have three dimensions, the last one corresponding to the scheduling variable v. For example, if the C-matrix corresponding to the first entry of v is the identity matrix, then C(:,:,1) = [1 0;0 1];.

Programmatic Use

Block Parameter: Ck

Type: character vector

Values: vector

Default: '[1 0]'

D-matrix — D-matrix of the state-space implementation

0.02 (default) | array | scalar

D-matrix of the state-space implementation. The D-matrix should have three dimensions, the last one corresponding to the scheduling variable v. For example, if the D-matrix corresponding to the first entry of v is the identity matrix, then D(:,:,1) = [1 0;0 1];.

Programmatic Use

Block Parameter: Dk

Type: character vector

Values: vector

Default: '0.02'

Initial state, x_initial — Initial states

0 (default) | vector

Initial states for the controller, that is, initial values for the state vector, z. It should have length equal to the size of the first dimension of A.

Programmatic Use

Block Parameter: x_initial

Type: character vector

Values: vector

Default: '0'

Poles of A-H*C — Desired poles

[-5 -2] (default) | vector

Desired poles of A-H*C, specified as a vector. Hence the number of pole locations defined should be equal to the dimension of the A-matrix.

Programmatic Use

Block Parameter: vec_w

Type: character vector

Values: vector

Default: '[-5 -2]'

More About

State-Space Controller

State-space controller implemented in both self-conditioned and standard state-space forms.

This Simulink^® model shows a state-space controller implemented in both self-conditioned and standard state-space forms. The actuator authority limits of ±0.5 units are modeled by the Saturation block.

Notice that the A-matrix has a zero in the 1,1 element, indicating integral action.

The top trace shows the conventional state-space implementation. The output of the controller winds up well past the actuator upper authority limit of +0.5. The lower trace shows that the self-conditioned form results in an actuator demand that tracks the upper authority limit, which means that when the sign of the control error, e, is reversed, the actuator demand responds immediately.

References

[1] Kautsky, Nichols, and Van Dooren, "Robust Pole Assignment in Linear State Feedback," International Journal of Control, Vol. 41, Number 5, 1985, pp. 1129-1155.

Extended Capabilities

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

Version History

Introduced before R2006a