Exercises - Optimal Control

Verification of a computed optimal trajectory
Suppose a direct-collocation solver returns the discrete solution
$\left\{ \boldsymbol{X}_k, \boldsymbol{U}_k \right\}_{k=0}^{K}$
(1)
for
$\min_{\boldsymbol{u}(\cdot)} \; J = \Phi\!\left(\boldsymbol{x}(t_f)\right) + \int_{t_0}^{t_f} L(\boldsymbol{x},\boldsymbol{u},t)\,\mathrm{d}t$
(2)
subject to
$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x},\boldsymbol{u},t).$
(3)
(a) Define a dynamic-defect residual on each mesh interval.
(b) Define the maximum endpoint-constraint residual.
(c) Define the maximum path-constraint violation.
(d) Explain how to re-simulate the reconstructed control using an independent differential-equation solver.
(e) Define a mesh-refinement table containing the objective, terminal residual, maximum defect, and maximum path violation.
(f) State what convergence behavior would support the claim that the numerical result is credible.
(g) Explain why a solver-success message alone is not evidence that the engineering conclusion is correct.
Complete derivation of the bilinear tangent steering law
Consider
$\dot{x}=u, \qquad \dot{y}=v, \qquad \dot{u}=a\cos\beta, \qquad \dot{v}=a\sin\beta,$
(4)
with
$x(0)=y(0)=u(0)=v(0)=0,$
(5)
$y(T)=H, \qquad v(T)=0,$
(6)
where $T$ is fixed. The terminal values $x(T)$ and $u(T)$ are free, and the objective is
$\max_{\beta(\cdot)}u(T).$
(7)
(a) Convert the problem to minimization form.
(b) Construct the Hamiltonian.
(c) Derive and integrate all four costate equations.
(d) Derive the interior stationarity condition
$\tan\beta = \frac{\lambda_v}{\lambda_u}.$
(8)
(e) Apply terminal transversality to show that
$\lambda_x(t)\equiv0, \qquad \lambda_u(t)\equiv-1.$
(9)
(f) Show that the general bilinear tangent law reduces to
$\tan\beta(t) = \tan\beta_0-ct.$
(10)
(g) Explain why the correct branch of $\beta$ must still be checked against the minimum principle.
Analytical state reconstruction for fixed-time ascent
Let
$q(t) = \tan\beta(t) = q_0-ct.$
(11)
(a) Show that
$\cos\beta = \frac{1}{\sqrt{1+q^2}}, \qquad \sin\beta = \frac{q}{\sqrt{1+q^2}}, \qquad \mathrm{d}t = -\frac{1}{c}\,\mathrm{d}q.$
(12)
(b) Starting from
$\dot{u}=a\cos\beta,$
(13)
derive
$u(t) = \frac{a}{c} \ln \left[ \frac{ q_0+\sqrt{1+q_0^2} }{ q(t)+\sqrt{1+q^2(t)} } \right].$
(14)
(c) Starting from
$\dot{v}=a\sin\beta,$
(15)
derive
$v(t) = \frac{a}{c} \left[ \sqrt{1+q_0^2} - \sqrt{1+q^2(t)} \right].$
(16)
(d) Use $v(T)=0$ to show that the nontrivial solution satisfies
$\beta(T)=-\beta_0.$
(17)
(e) Derive
$c = \frac{2\tan\beta_0}{T}.$
(18)
(f) Show that
$\tan\beta(t) = \tan\beta_0 \left( 1-\frac{2t}{T} \right).$
(19)
(g) Explain the symmetry of the steering law about $t=T/2$ .
Altitude equation and terminal horizontal velocity
For the fixed-time ascent problem, the analytical vertical position is
$y(t) = \frac{a}{2c^2} \left[ \bigl( \tan\beta_0-\tan\beta \bigr) \bigl( \sec\beta_0+\sec\beta \bigr) - \ln \left( \frac{ \tan\beta_0+\sec\beta_0 }{ \tan\beta+\sec\beta } \right) \right].$
(20)
(a) Substitute
$\beta(T)=-\beta_0, \qquad c=\frac{2\tan\beta_0}{T}$
(21)
and derive the scalar altitude equation
$\frac{4H}{aT^2} = \frac{ 2\tan\beta_0\sec\beta_0 - \ln \left( \frac{ \sec\beta_0+\tan\beta_0 }{ \sec\beta_0-\tan\beta_0 } \right) }{ 2\tan^2\beta_0 }.$
(22)
(b) Explain why the original trajectory-optimization problem has now been reduced to one scalar nonlinear equation.
(c) Derive the expression for the terminal horizontal velocity $u(T)$ .
(d) Describe a numerical procedure for solving for $\beta_0$ .
(e) State the endpoint and minimum-principle checks that must be performed after a root is obtained.
Necessary conditions for fuel-optimal lunar landing
Consider
$\dot{h}=v, \qquad \dot{v}=-g+u,$
(23)
with
$0\leq u\leq3,$
(24)
$h(0)=h_0, \qquad v(0)=v_0,$
(25)
$h(t_f)=0, \qquad v(t_f)=0,$
(26)
and free $t_f$ . Minimize
$J = \int_0^{t_f}u(t)\,\mathrm{d}t.$
(27)
Use the chapter’s normal Hamiltonian convention
$H = 1+\lambda_hv+\lambda_v(-g+u).$
(28)
(a) Derive the costate equations and their general solutions.
(b) Define the switching function
$\sigma(t) = 1+\lambda_v(t).$
(29)
(c) Derive the bang–bang control law.
(d) Prove that a nondegenerate singular arc cannot exist.
(e) Show that the optimal control has at most one switching time.
(f) Derive the free-final-time condition
$H(t_f)=0.$
(30)
Piecewise analytical lunar-landing trajectory
For the lunar-landing model, first derive the state solution on an interval $[t_a,t]$ when the initial values at $t_a$ are $h_a$ and $v_a$ .
(a) For $u=0$ , derive $v(t)$ and $h(t)$ .
(b) For $u=3$ , derive $v(t)$ and $h(t)$ .
(c) Assume a coast-then-burn structure,
$u(t) = \begin{cases} 0, & 0\leq t<t_s,\\ 3, & t_s\leq t\leq t_f. \end{cases}$
(31)
Derive $h(t_s)$ and $v(t_s)$ .
(d) Enforce continuity and derive the two algebraic equations produced by
$h(t_f)=0, \qquad v(t_f)=0.$
(32)
(e) Express the fuel cost in terms of $t_s$ and $t_f$ .
(f) Repeat the formulation for a burn-then-coast structure.
(g) State how the admissible structures should be compared and verified.
Direct and indirect verification of lunar landing
Formulate a numerical verification study for the lunar-landing problem.
(a) Write a direct-shooting formulation using $t_s$ and $t_f$ as decision variables for a coast-then-burn control.
(b) Write a direct-collocation formulation with state variables $h$ and $v$ , control $u$ , and free final time.
(c) State the boundary and control constraints.
(d) Define the terminal residuals.
(e) Define the Hamiltonian residual at the final time.
(f) Explain how to compare the shooting and collocation solutions.
(g) Explain why a smooth numerical approximation of the switch does not by itself disprove the bang–bang analytical structure.
Total specific energy and constant-energy contours
Consider the longitudinal aircraft equations
$\dot{V} = \frac{T\cos\alpha-D}{m} - g\sin\gamma, \qquad \dot{h} = V\sin\gamma.$
(33)
Define
$E = \frac{V^2}{2}+gh.$
(34)
(a) Differentiate $E$ with respect to time.
(b) Show explicitly that the gravitational terms cancel.
(c) Derive
$\dot{E} = \frac{V}{m} \left( T\cos\alpha-D \right).$
(35)
(d) Under the small-angle approximation, derive
$\dot{E} \approx \frac{V}{m}(T-D).$
(36)
(e) For constant $E$ , derive
$h = \frac{E}{g} - \frac{V^2}{2g}.$
(37)
(f) Derive
$\frac{\mathrm{d}h}{\mathrm{d}V} = -\frac{V}{g}.$
(38)
(g) Derive the altitude loss required to accelerate from $V_1$ to $V_2$ at constant energy.
Energy-state minimum-time optimality conditions
Consider the reduced model
$\dot{E} = P_s(E,V) = \frac{V}{m} \left[ T(E,V)-D(E,V) \right],$
(39)
with
$\min t_f = \min \int_0^{t_f}1\,\mathrm{d}t.$
(40)
(a) Rewrite the final time as
$t_f = \int_{E_0}^{E_f} \frac{1}{P_s(E,V)}\,\mathrm{d}E.$
(41)
(b) Explain directly from this integral why minimum time requires maximum specific excess power at each energy level.
(c) Construct the reduced Hamiltonian
$H = 1+\lambda_EP_s(E,V).$
(42)
(d) Derive the stationarity condition
$\frac{\partial P_s}{\partial V}=0.$
(43)
(e) Derive the costate equation and show that $\lambda_E$ cannot change sign along a normal extremal.
(f) Use the free-final-time condition to show that $\lambda_E<0$ for a feasible climb with $P_s>0$ .
(g) Show that the speed schedule satisfies
$T-D + V \left( T_V-D_V \right) = 0.$
(44)
(h) State the second-derivative condition required for a local maximum of specific excess power.
Full-model refinement of the energy-state solution
Let the full aircraft state and control be
$\boldsymbol{x} = \begin{bmatrix} V&\gamma&h&x&m \end{bmatrix}^{\mathsf T}, \qquad \boldsymbol{u} = \begin{bmatrix} \alpha&\eta \end{bmatrix}^{\mathsf T}.$
(45)
The objective is to minimize $t_f$ .
(a) Write the full longitudinal dynamics used in the chapter.
(b) Write the minimum-time Hamiltonian for an indirect formulation.
(c) State the interior stationarity equations with respect to $\alpha$ and $\eta$ .
(d) Write the normalized-time dynamics on $\tau\in[0,1]$ .
(e) List the flight-envelope constraints discussed in the chapter.
(f) Explain how an energy-state speed schedule can be converted into an initial guess for the full problem.
(g) Define verification checks for terminal feasibility, path feasibility, mesh convergence, Hamiltonian consistency, and independent re-simulation.
(h) Explain why agreement in qualitative dive–climb structure is useful but does not prove quantitative accuracy of the reduced model.