state_space – DYNAMICS

public interface state_space

Module Procedures

state_space_init state_space_init_scalar state_space_init_matrices state_space_init_pid state_space_init_pid_plant

Module Procedures

private pure function state_space_init(m, b, k, n_out) result(rst)

Initializes the state space model.

The output matrix $C$ is initialized to one, and the feedthrough matrix $D$ is initialized to zero.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in),		dimension(:,:)	::	m	The N-by-N mass matrix.
real(kind=real64),	intent(in),		dimension(size(m, 1), size(m, 2))	::	b	The N-by-N damping matrix.
real(kind=real64),	intent(in),		dimension(size(m, 1), size(m, 2))	::	k	The N-by-N stiffness matrix.
integer(kind=int32),	intent(in),	optional		::	n_out	The number of outputs. The default is 1.

Return Value type(state_space)

The [[state_space]] model.

private pure function state_space_init_scalar(m, b, k) result(rst)

Initializes the state space model.

The output matrix $C$ is initialized to one, and the feedthrough matrix $D$ is initialized to zero.

Arguments

Type	Intent		Name
real(kind=real64),	intent(in)	::	m	The mass.
real(kind=real64),	intent(in)	::	b	The damping.
real(kind=real64),	intent(in)	::	k	The stiffness.

Return Value type(state_space)

The [[state_space]] model.

private pure function state_space_init_matrices(a, b, c, d) result(rst)

Initializes the state space model.

Arguments

Type	Intent	Attributes		Name
real(kind=real64),	intent(in),	dimension(:,:)	::	a	The N-by-N dynamics matrix.
real(kind=real64),	intent(in),	dimension(:,:)	::	b	The N-by-M input matrix.
real(kind=real64),	intent(in),	dimension(:,:)	::	c	The P-by-N output matrix.
real(kind=real64),	intent(in),	dimension(:,:)	::	d	The P-by-M feedthrough matrix.

Return Value type(state_space)

The resulting [[state_space]] object.

private pure function state_space_init_pid(kp, ki, kd, tau, a, b, c, d) result(rst)

Initializes a state-space model that employs a closed-loop PID controller.

The PID model is augmented into the plant model as follows.

$e = r - y$ $\dot{x_1} = e$ $\dot{x_3} = -\frac{x_3}{\tau} + \frac{e}{\tau}$ $u = K_{i} x_{i} + K_{p} e + \frac{K_{d}}{\tau} \left(e - x_{3} \right)$ $\alpha = K_{p} + \frac{K_{d}}{\tau}$ $\beta = \frac{1}{1 + \alpha D}$ $x = \left[ \begin{matrix} x_{p} \\ x_{1} \\ x_{3} \end{matrix} \right]$ $\dot{x} = A_{cl} x + B_{cl} r$ $y = C_{cl} x + D_{cl} r$

Where the augmented matrices are as follows.

$A_{cl} = \left[ \begin{matrix} A - \alpha \beta B C & \beta K_{i} B & -\frac{\beta K_{d}}{\tau} B \\ -C + \alpha \beta D C & -\beta K_{i} D & \frac{\beta K_{d}}{\tau} D \\ \frac{1}{\tau}\left(-C + \alpha \beta D C \right) & -\frac{\beta K_{i}}{\tau} D & \frac{1}{\tau} \left( 1 + \frac{\beta K_{d}}{\tau^{2}} D \right) \end{matrix} \right]$ $B_{cl} = \left[ \begin{matrix} \alpha \beta B \\ 1 - \alpha \beta D \\ \frac{1}{\tau} \left( 1 - \alpha \beta D \right) \end{matrix} \right]$ $C_{cl} = \left[ \begin{matrix} C - \alpha \beta D C & \beta K_{i} D & -\frac{\beta K_{d}}{\tau} D \end{matrix} \right]$ $D_{cl} = \alpha \beta D$

Arguments

Type	Intent	Attributes		Name
real(kind=real64),	intent(in)		::	kp	The proportional gain term.
real(kind=real64),	intent(in)		::	ki	The integral gain term.
real(kind=real64),	intent(in)		::	kd	The derivative gain term.
real(kind=real64),	intent(in)		::	tau	The time constant of the first order derivative filter $K_{d} \frac{s}{\tau s + 1}$ .
real(kind=real64),	intent(in),	dimension(:,:)	::	a	The N-by-N dynamics matrix for the plant.
real(kind=real64),	intent(in),	dimension(size(a, 1), 1)	::	b	The N-by-1 input matrix for the plant.
real(kind=real64),	intent(in),	dimension(1, size(a, 1))	::	c	The 1-by-N output matrix for the plant.
real(kind=real64),	intent(in),	dimension(1, 1)	::	d	The 1-by-1 feedthrough matrix for the plant.

Return Value type(state_space)

The resulting [[state_space]] object.

private pure function state_space_init_pid_plant(kp, ki, kd, tau, plant) result(rst)

Initializes a state-space model that employs a closed-loop PID controller.

The PID model is augmented into the plant model as follows.

Where the augmented matrices are as follows.

Arguments

Type	Intent		Name
real(kind=real64),	intent(in)	::	kp	The proportional gain term.
real(kind=real64),	intent(in)	::	ki	The integral gain term.
real(kind=real64),	intent(in)	::	kd	The derivative gain term.
real(kind=real64),	intent(in)	::	tau	The time constant of the first order derivative filter $K_{d} \frac{s}{\tau s + 1}$ .
class(state_space),	intent(in)	::	plant	The plant model.

Return Value type(state_space)

The resulting [[state_space]] object.

state_space Interface

Contents

Module Procedures

public interface state_space

Contents

Module Procedures

Module Procedures

private pure function state_space_init(m, b, k, n_out) result(rst)

Arguments

Return Value type(state_space)

private pure function state_space_init_scalar(m, b, k) result(rst)

Arguments

Return Value type(state_space)

private pure function state_space_init_matrices(a, b, c, d) result(rst)

Arguments

Return Value type(state_space)

private pure function state_space_init_pid(kp, ki, kd, tau, a, b, c, d) result(rst)

Arguments

Return Value type(state_space)

private pure function state_space_init_pid_plant(kp, ki, kd, tau, plant) result(rst)

Arguments

Return Value type(state_space)