NLOptControl.jl¶

Definition¶

• given \((N+1)\) unique data points
  • \((x_0,y_0),(x_1,y_1),....,(x_N,y_N)\)
• we can create an \(N^{th}\) order Lagrange interpolating polynomial

where,

\[\begin{eqnarray} f(x_0) = y_0\\ f(x_1) = y_1\\ .\\ .\\ f(x_i) = y_i\\ .\\ f(x_N) = y_N \end{eqnarray}\]

So, we are just multiplying by the given \(y_i\) values.

Lagrange Basis Polynomials¶

More information on Lagrange Basis Polynomials is here

so expanding this,

\[\begin{eqnarray} \mathcal{L}_i(x) &=\frac{x-x_0}{x_i-x_0}\frac{x-x_1}{x_i-x_1}...\\ &...\frac{x-x_{i-1}}{x_i-x_{i-1}}...\\ &...\frac{x-x_{i+1}}{x_i-x_{i+1}}...\\ &...\frac{x-x_N}{x_i-x_N} \end{eqnarray}\]

Notice that we do not include the term where \(i==j\)!

Please see lpf for details on implementation.

Euler Method¶

Trapezoidal Method¶

Change of Interval¶

To can change the limits of the integration (in order to apply Quadrature), we introduce \(\tau \in [-1,+1]\) as a new independent variable and perform a change of variable for \(t\) in terms of \(\tau\), by defining:

\[\tau = \frac{2}{t_{{N}_{t}}-t_0}t - \frac{t_{N_t}+t_0}{t_{N_t}-t_0}\]

Polynomial Interpolation¶

Select a set of \(N_t+1\) node points:

\[\mathbf{\tau} = [\tau_0,\tau_1,\tau_2,.....,\tau_{N_t}]\]

• These none points are just numbers
  • Increasing and distinct numbers \(\in [-1,+1]\)

A unique polynomial \(P(\tau)\) exists (i.e. \(\exists! P(\tau)\)) of a maximum degree of \(N_t\) where:

\[f(\tau_k)=P(\tau_k),\;\;\;k={0,1,2,....N_t}\]

• So, the function evaluated at \(\tau_k\) is equivalent the this polynomial evaluated at that point.

But, between the intervals, we must approximate \(f(\tau)\) as:

\[f(\tau) \approx P(\tau)= \sum_{k=0}^{N_t}f(\tau_k)\phi_k(\tau)\]

with \(\phi_k(•)\) are basis polynomials that are built by interpolating \(f(\tau)\) at the node points.

Approximating Derivatives¶

We can also approximate the derivative of a function \(f(\tau)\) as:

With \(\mathbf{D}\) is a \((N_t+1)\times(N_t+1)\) differentiation matrix that depends on:

• values of \(\tau\)
• type of interpolating polynomial

Now we have an approximation of \(\dot{f}(\tau_k)\) that depends only on \(f(\tau)\)!

Approximating Integrals¶

The integral we are interested in evaluating is:

This can be approximated using quadrature:

where \(w_k\) are quadrature weights and depend only on:

• values of \(\tau\)
• type of interpolating polynomial

Legendre Pseudospectral Method¶

• Polynomial

Define an N order Legendre polynomial as:

\[L_N(\tau) = \frac{1}{2^NN!}\frac{\mathrm{d}^n}{\mathrm{d}\tau^N}(\tau^2-1)^N\]

• Nodes

• Differentiation Matrix
• Interpolating Polynomial Function

Single Phase Optimal Control¶

Find:

• The state: \(\mathbf{x}(t)\)
• The control: \(\mathbf{u}(t)\)
• The integrals: \(\mathbf{q}\)
• The initial time: \(t_0\)
• The final time: \(t_f\)

To Minimize:

\[J = \Phi(\mathbf{x}(t_0),\mathbf{x}(t_f),\mathbf{q},t_0,t_f)\]

That Satisfy the Following Constraints:

• Dynamic Constraints:

\[\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} = \mathbf{\psi}(\mathbf{x}(t),\mathbf{u}(t),t)\]

• Inequality Path Constraints:

\[\begin{split}\mathbf{c}_{min} <= \mathbf{c}(\mathbf{x}(t),\mathbf{u}(t),t) <= \mathbf{c}_{max}\end{split}\]

• Integral Constraints:

\[q_i = \int_{t_0}^{t_f} \Upsilon_i(\mathbf{x}(t),\mathbf{u}(t),t)\, \mathrm{d}t,\;\;(i=1,....,n_q)\]

• Event Constraints:

\[\begin{split}\mathbf{b}_{min} <= \mathbf{b}(\mathbf{x}(t_0),\mathbf{x}(t_f),t_f,\mathbf{q}) <= \mathbf{b}_{max}\end{split}\]

Change of Interval¶

To can change the limits of the integration (in order to apply Quadrature), we introduce \(\tau \in [-1,+1]\) as a new independent variable and perform a change of variable for \(t\) in terms of \(\tau\), by defining:

\[t = \frac{t_f - t_0}{2}\tau + \frac{t_f + t_0}{2}\]

The optimal control problem defined above (TODO: figure out equation references), is now redefined in terms of \(\tau\) as:

Find:

• The state: \(\mathbf{x}(\tau)\)
• The control: \(\mathbf{u}(\tau)\)
• The integrals: \(\mathbf{q}\)
• The initial time: \(t_0\)
• The final time: \(t_f\)

To Minimize:

\[J = \Phi(\mathbf{x}(-1),\mathbf{x}(+1),\mathbf{q},t_0,t_f)\]

That Satisfy the Following Constraints:

• Dynamic Constraints:

\[\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}\tau} = \frac{t_f-t_0}{2} \mathbf{\psi}(\mathbf{x}(\tau),\mathbf{u}(\tau),\tau,t_0,t_f)\]

• Inequality Path Constraints:

\[\begin{split}\mathbf{c}_{min} <= \mathbf{c}(\mathbf{x}(\tau),\mathbf{u}(\tau),\tau,t_0,t_f) <= \mathbf{c}_{max}\end{split}\]

• Integral Constraints:

\[q_i = \frac{t_f-t_0}{2} \int_{-1}^{+1} \Upsilon_i(\mathbf{x}(\tau),\mathbf{u}(\tau),\tau,t_0,t_f)\, \mathrm{d}\tau,\;\;(i=1,....,n_q)\]

• Event Constraints:

\[\begin{split}\mathbf{b}_{min} <= \mathbf{b}(\mathbf{x}(-1),\mathbf{x}(+1),t_f,\mathbf{q}) <= \mathbf{b}_{max}\end{split}\]

Divide The Interval \(\tau \in [-1,+1]\)¶

The interval \(\tau \in [-1,+1]\) is now divided into a mesh of K mesh intervals as:

\[[T_{k-1},T_k], k = 1,...,T_K\]

with \((T_0,...,T_K)\) being the mesh points; which satisfy:

\[\begin{split}-1 = T_0 < T_1 < T_2 < T_3 < ........... < T_{K-1} < T_K = T_f = +1\end{split}\]

Rewrite the Optimal Control Problem using the Mesh¶

Find:

• The state : \(\mathbf{x}^{(k)}(\tau)\) in mesh interval k
• The control: \(\mathbf{u}^{(k)}(\tau)\) in mesh interval k
• The integrals: \(\mathbf{q}\)
• The initial time: \(t_0\)
• The final time: \(t_f\)

To Minimize:

\[J = \Phi(\mathbf{x}^{(1)}(-1),\mathbf{x}^{(K)}(+1),\mathbf{q},t_0,t_f)\]

That Satisfy the Following Constraints:

• Dynamic Constraints:

\[\frac{\mathrm{d}\mathbf{x}^{(k)}(\tau^{(k)})}{\mathrm{d}\tau^{(k)}} = \frac{t_f-t_0}{2} \mathbf{\psi}(\mathbf{x}^{(k)}(\tau^{(k)}),\mathbf{u}^{(k)}(\tau^{(k)}),\tau^{(k)},t_0,t_f),\;\;(k=1,...,K)\]

• Inequality Path Constraints:

• Integral Constraints:

• Event Constraints:

• State Continuity

  • Also, we must now constrain the state to be continuous at each interior mesh point \((T_1,...T_{k-1})\) by enforcing:

    \[\mathbf{y}^{k}(T_k) = \mathbf{y}^{k+1}(T_k)\]

Optimal Control Problem Approximation¶

The optimal control problem will now be approximated using the Radau Collocation Method as which follows the description provided by [BGar11]. In collocation methods, the state and control are discretized at particular points within the selected time interval. Once this is done the problem can be transcribed into a nonlinear programming problem (NLP) and solved using standard solvers for these types of problems, such as IPOPT or KNITRO.

For each mesh interval \(k\in[1,..,K]\):

\[\begin{eqnarray} \mathbf{x}^{(k)}(\tau)&\approx\mathbf{X}^{(k)}(\tau)=\displaystyle\sum_{j=1}^{N_k+1}\mathbf{X}_j^{(k)}\frac{\mathrm{d}\mathcal{L}_j^{k}(\tau)}{\mathrm{d}\tau}\\ where,\;\;&\\ \mathcal{L}_j^{k}(\tau)&=\prod_{\substack{l=1 \\ l\neq j}}^{N_k+1}\frac{\tau-\tau_l^{(k)}}{\tau_j^{(k)}-\tau_l^{(k)}}\\ and,\;\;&\\ &D_{ki}=\dot{\mathcal{L}}_i(\tau_k)=\frac{\mathrm{d}\mathcal{L}_j^{k}(\tau)}{\mathrm{d}\tau} \end{eqnarray}\]

also,

• \(\mathcal{L}_j^{(k)}(\tau),\;\;(j=1,...,N_k+1)\) is a basis of Lagrange polynomials

• \((\tau_1^{k},.....,\tau_{N_k}^{(k)})\) are the Legendre-Gauss-Radau collocation points in mesh interval k

  • defined on the subinterval \(\tau^{(k)}\in[T_{k-1},T_k]\)
  • \(\tau_{N_k+1}^{(k)}=T_k\) is a noncollocated point

A basic description of Lagrange Polynomials is presented in Lagrange Interpolating Polynomials

The \(\mathbf{D}\) matrix:

• Has a size \(= [N_c]\times[N_c+1]\)

  • with \((1<=k<=N_c), (1<=i<=N_c+1)\)

  • this non-square shape because the state approximation uses the \(N_c+1\) points:

    \((\tau_1,...\tau_{N_c+1})\)

  • but collocation is only done at the \(N_c\) LGR points:

    \((\tau_1,...\tau_{N_c})\)

If we define the state matrix as:

The dynamics are collocated at the \(N_c\) LGR points using:

\(\mathbf{D}_k\mathbf{X}^{LGR} = \frac{(t_f-t_0)}{2}\mathbf{f}(\mathbf{X}_k,\mathbf{U}_k,\tau,t_0,t_f)\;\;for\;\;k = {1,...,Nc}\)

with,

• \(\mathbf{D}_k\) being the \(k^{th}\) row of the \(\mathbf{D}\) matrix.

References