7 Introduction to Optimisation
251
state variables at intermediate grid points are now control variables to be optimised.
Hence, the number of control parameters in y increases with respect to the single
shooting method, precisely n y = (m − 1)(n s + n c · n p ), where n s is the number of
state variables, n c the control components and n p the control parameters per each
component. The defect equations can be expressed in the general form as:
c(y) =
⎛
⎜
⎝
x
p
2 − x 2
. . .
x
p
f − x f
⎞
⎟
⎠
(7.13)
where again the goal is to nullify c(y). The dimension that the NLP subproblem
shall solve, in order to link the different phases and minimise the objective function,
dramatically increases with an increasing number of steps. However, the effects
of changing a particular parameter are more intuitive for smaller steps, leading
to an improvement of the convergence properties. In addition, the main drawback
of the single shooting is solved, and, when the number of steps is high enough,
the variables related to the first stages of the trajectory do not heavily influence
the last phases. The segment decoupling mathematically translates into very sparse
Jacobian and Hessian matrices, later involved by the NLP algorithm. For example,
the Jacobian gets sparser and sparser as more phases are employed, because
the percentage of non-zero elements is proportional to 1/(m − 1). This sparsity
can be exploited to construct a computationally efficient non-linear programming
subroutine, making the multiple shooting method both robust and competitive [77].
Collocation
The basic goal of collocation methods is to avoid repeated propagations over
each segment. This is achieved by partitioning again the whole trajectory into
m − 1 segments, leading to m grid points. Hence, the trajectory is only represented
by the set of state variables x(t k ) and their derivatives f(t k , x(t k ), u(t k )) at mesh
points as well as the control profile nodes u(t k ). As these values are treated
as NLP variables, gathered in the vector y, the optimal control problem has
been completely transcribed into a finite-dimensional NLP. For this reason, also
collocation methods need a first guess solution, which can be sought with the
aforementioned approaches. The state, state-derivative and control values within
each interval are computed by interpolation through piecewise functions, usually
Hermite (third order), Chebyshev or Lagrange polynomials (see [68] for detailed
schemes) or Fourier series [78], whose coefficients depend on the adjacent grid
points’ state and derivatives. This a priori shape replaces the numerical integration
process of shooting techniques with a much faster analytical propagation.
The differential equations ˙
x = f(t, x(t), u(t)) are substituted by a discretised
form, which for a simple Euler scheme takes the following form:
251
state variables at intermediate grid points are now control variables to be optimised.
Hence, the number of control parameters in y increases with respect to the single
shooting method, precisely n y = (m − 1)(n s + n c · n p ), where n s is the number of
state variables, n c the control components and n p the control parameters per each
component. The defect equations can be expressed in the general form as:
c(y) =
⎛
⎜
⎝
x
p
2 − x 2
. . .
x
p
f − x f
⎞
⎟
⎠
(7.13)
where again the goal is to nullify c(y). The dimension that the NLP subproblem
shall solve, in order to link the different phases and minimise the objective function,
dramatically increases with an increasing number of steps. However, the effects
of changing a particular parameter are more intuitive for smaller steps, leading
to an improvement of the convergence properties. In addition, the main drawback
of the single shooting is solved, and, when the number of steps is high enough,
the variables related to the first stages of the trajectory do not heavily influence
the last phases. The segment decoupling mathematically translates into very sparse
Jacobian and Hessian matrices, later involved by the NLP algorithm. For example,
the Jacobian gets sparser and sparser as more phases are employed, because
the percentage of non-zero elements is proportional to 1/(m − 1). This sparsity
can be exploited to construct a computationally efficient non-linear programming
subroutine, making the multiple shooting method both robust and competitive [77].
Collocation
The basic goal of collocation methods is to avoid repeated propagations over
each segment. This is achieved by partitioning again the whole trajectory into
m − 1 segments, leading to m grid points. Hence, the trajectory is only represented
by the set of state variables x(t k ) and their derivatives f(t k , x(t k ), u(t k )) at mesh
points as well as the control profile nodes u(t k ). As these values are treated
as NLP variables, gathered in the vector y, the optimal control problem has
been completely transcribed into a finite-dimensional NLP. For this reason, also
collocation methods need a first guess solution, which can be sought with the
aforementioned approaches. The state, state-derivative and control values within
each interval are computed by interpolation through piecewise functions, usually
Hermite (third order), Chebyshev or Lagrange polynomials (see [68] for detailed
schemes) or Fourier series [78], whose coefficients depend on the adjacent grid
points’ state and derivatives. This a priori shape replaces the numerical integration
process of shooting techniques with a much faster analytical propagation.
The differential equations ˙
x = f(t, x(t), u(t)) are substituted by a discretised
form, which for a simple Euler scheme takes the following form: