state_space Derived Type

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

Initializes the state space model. For the second-order mechanical system $$M\ddot{q}+B\dot{q}+Kq=u,$$ the state $x=[q^T\ \dot{q}^T]^T$ satisfies $$\dot{x}=Ax+Bu,\qquad y=Cx+Du.$$

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

Arguments

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 scalar realization corresponds to $$m\ddot{q}+b\dot{q}+kq=u,\qquad A=\begin{bmatrix}0&1\\-k/m&-b/m\end{bmatrix},\quad B=\begin{bmatrix}0\\1/m\end{bmatrix}.$$

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

Arguments

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. The stored realization uses the continuous-time equations $$\dot{x}=Ax+Bu,\qquad y=Cx+Du.$$

Arguments

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.

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

Arguments

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.

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

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

Return Value type(state_space)

The resulting [[state_space]] object.

private pure function ss_eval_deriv(this, u, x) result(rst)

Evaluates the state time derivative $\dot{x}(t) = A x(t) + B u(t)$. This is the vector field evaluated at one state and input sample.

Arguments

Return Value real(kind=real64), allocatable, dimension(:)

The N-element state time derivative vector.

private pure function ss_eval_output(this, u, x) result(rst)

Evaluates the output vector $y(t) = C x(t) + D u(t)$. The $D u$ term is the direct feedthrough contribution.

Arguments

Return Value real(kind=real64), allocatable, dimension(:)

The P-element output array.

private pure function ss_poles(this) result(rst)

Computes the poles of the state space model. The poles are the eigenvalues of $A$ and are the roots of $$\det(sI-A)=0.$$

Arguments

Return Value complex(kind=real64), allocatable, dimension(:)

The poles of the model.

private pure function ss_transfer_fcn(this, s) result(rst)

Evaluates the transfer functions for the model at the parameter $s$. In the Laplace domain, $$H(s)=C(sI-A)^{-1}B+D.$$

Arguments

Return Value complex(kind=real64), allocatable, dimension(:,:)

The resulting transfer functions.

private pure function ss_transfer_fcn_omega(this, omega) result(rst)

Evaluates the transfer functions for the model at frequency $\omega$. This evaluates the transfer matrix on the imaginary axis, $s=j\omega$.

Arguments

Return Value complex(kind=real64), allocatable, dimension(:,:)

The resulting transfer functions.

private pure function ss_transfer_fcn_array(this, s) result(rst)

Evaluates the transfer functions for the model at the frequencies given in the array $s$.

Arguments

Return Value complex(kind=real64), allocatable, dimension(:,:,:)

The resulting transfer functions, with each page of the array containing the transfer functions for a specific frequency.

private pure function ss_transfer_fcn_omega_array(this, omega) result(rst)

Evaluates the transfer functions for the model at the frequencies given in the array $omega$.

Arguments

Return Value complex(kind=real64), allocatable, dimension(:,:,:)

The resulting transfer functions, with each page of the array containing the transfer functions for a specific frequency.

private pure function ss_zeros(this) result(rst)

Computes the zeros of the state space model. Transmission zeros are obtained from the generalized eigenproblem of the Rosenbrock system matrix $$\mathcal{R}(s)=\begin{bmatrix}sI-A&-B\\C&D\end{bmatrix}.$$

Arguments

Return Value complex(kind=real64), allocatable, dimension(:)

The zeros of the model.