Aerospace and Electronic Systems Magazine April 2018 - 7

Wahl and Turkoglu
Here, a linear differential equation is produced based on the
previous equations:
∂
∂τ

 xt* − xτ*   A
=
 *
*
λt − λτ   −C

− B   xt* − xτ* 


− AT  λt* − λτ* 

(12)

where A = f x − f u H uu−1H ux, B = f u H uu−1 f uT , and C = H xx − H xu H uu−1H ux.
pt and pτ are canceled.
The derivative of the nonlinear function F with respect to time
is given by the following:

Because a real-life applicable strategy is desired, the control
system being investigated does not involve any form of linearization but rather contains nonlinear terms in its equations of
motion.
The following algorithm is used by the computer program to
simulate the control system:
C

Initialize and set up.

C

Determine the first time horizon.

C

While the simulation is running:
Read the current plant conditions from the appropriate
subsystem or subsystems.

dF
= λt* (T , t ) − ϕ xx xt* (T , t ) − ϕ xp pt ( t + T )
dt
+ λτ* (T , t ) − ϕ xx xτ* (T , t ) − ϕ xp pτ ( t + T ) 

dT
dt

(13)

Substitute state information into various system equations, and differentiate as necessary.

The relationship between the costate and other variables is expressed as follows:

λt* − λτ* = S (τ , t ) ( xt* − xτ* ) + c (τ , t )

Integrate system equations forward over the time horizon
to get costate values (which in turn determine the optimum outcome).

(14)

Integrate system equations backward to get the sequence
of necessary control efforts to achieve the optimum outcome.

where
Sτ = − AT S − SA + SBS − C

(

)

cτ = − AT − SB c

Wait until the current time cycle is complete, and then determine the new time horizon.

The following conditions must hold:
S (T , t ) = ϕ xx

τ =T

(

, c (T , t ) = H xT + ϕ xx f + ϕ xp pt

)

τ =T

 dT 
1 + t  + As F
d 

(16)

Therefore, the differential equation of λ(t) to be integrated in real
time is obtained as
dλ ( t )
dt

= − H xT + c ( 0, t ) .

(17)

The basic idea of the backward sweep method is to integrate (7)
forward and then integrate (15) backward along the τ axis at each
time t. Then, the differential equation of λ(t) is integrated for one
step forward along the t axis so as to determine the optimal control
effort from (9).

SOLUTION STRATEGIES
The solution investigated by this research involves the use of a
real-time nonlinear receding horizon controller (NRHC). One advantage of receding horizon controller schemes is that they can be
made to optimize for some balance of control effort and precision
of final outcome, which is specified by the use of a cost function.
Furthermore, they can be made to take into account limitations of
the system and constraints on the final outcome in the process of
determining the optimal solution.
APRIL 2018

Send the first control effort in the sequence to the actuator
subsystem or subsystems.

(15)

C

Loop until the objective is met.

C

When simulation is done, plot the results.

SIMULATION RESULTS
In the process of investigating the proposed nonlinear receding
time-horizon control, a version of Ohtsuka's approach [16] was
implemented for this solution. Starting with the equations of motion specified by Lu [15], derivations were conducted to solve the
Euler-Lagrange equations symbolically (as specified in [11]) and
determine the Hamiltonian. An iterative routine was written to perform both forward and backward integrations, integrate forward
one more time to determine the desired state and costate for the
next iteration, and finally apply the determined control effort to
achieve that state.
To test the effectiveness of the resulting system, a variety of
trajectories of desired angles of attack were used as sample cases.
To be considered a successful system, it was necessary to be able to
reasonably track these various reference trajectories.

PREPLANNED TRAJECTORIES
In a typical launch vehicle, the overall trajectory of ascent is
planned before the vehicle is launched. Therefore, testing of
some reference trajectories was performed as if the trajectory
had been determined ahead of time, with the intention of observing the optimized path that resulted. Reference trajectories

IEEE A&E SYSTEMS MAGAZINE

7



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