Control

A simple control problem on a system usually involves a variable $x(t)$ that denotes the state of the system over time, and a variable $u(t)$ that denotes the input into the system over time. Linear constraints are used to capture the evolution of the system over time:

\[x(t) = Ax(t - 1) + Bu(t), \ \text{for} \ t = 1,\ldots, T,\]

where the numerical matrices $A$ and $B$ are called the dynamics and input matrices, respectively.

The goal of the control problem is to find a sequence of inputs $u(t)$ that will allow the state $x(t)$ to achieve specified values at certain times. For example, we can specify initial and final states of the system:

\[\begin{aligned} x(0) &= x_i \\ x(T) &= x_f \end{aligned}\]

Additional states between the initial and final states can also be specified. These are known as waypoint constraints. Often, the input and state of the system will have physical meaning, so we often want to find a sequence inputs that also minimizes a least squares objective like the following:

\[\sum_{t = 0}^T \|Fx(t)\|^2_2 + \sum_{t = 1}^T\|Gu(t)\|^2_2,\]

where $F$ and $G$ are numerical matrices.

We'll now apply the basic format of the control problem to an example of controlling the motion of an object in a fluid over $T$ intervals, each of $h$ seconds. The state of the system at time interval $t$ will be given by the position and the velocity of the object, denoted $p(t)$ and $v(t)$, while the input will be forces applied to the object, denoted by $f(t)$. By the basic laws of physics, the relationship between force, velocity, and position must satisfy:

\[ \begin{aligned} p(t+1) &= p(t) + h v(t) \\ v(t+1) &= v(t) + h a(t) \end{aligned}.\]

Here, $a(t)$ denotes the acceleration at time $t$, for which we use $a(t) = f(t) / m + g - d v(t)$, where $m$, $d$, $g$ are constants for the mass of the object, the drag coefficient of the fluid, and the acceleration from gravity, respectively.

Additionally, we have our initial/final position/velocity conditions:

\[ \begin{aligned} p(1) &= p_i\\ v(1) &= v_i\\ p(T+1) &= p_f\\ v(T+1) &= 0 \end{aligned}\]

One reasonable objective to minimize would be

\[ \text{objective} = \mu \sum_{t = 1}^{T+1} (v(t))^2 + \sum_{t = 1}^T (f(t))^2\]

We would like to keep both the forces small to perhaps save fuel, and keep the velocities small for safety concerns. Here $\mu$ serves as a parameter to control which part of the objective we deem more important, keeping the velocity small or keeping the force small.

The following code builds and solves our control example:

using Convex, SCS, Plots

# Some constraints on our motion
# The object should start from the origin, and end at rest
initial_velocity = [-20; 100]
final_position = [100; 100]

T = 100 # The number of timesteps
h = 0.1 # The time between time intervals
mass = 1 # Mass of object
drag = 0.1 # Drag on object
g = [0, -9.8] # Gravity on object

# Declare the variables we need
position = Variable(2, T)
velocity = Variable(2, T)
force = Variable(2, T - 1)

# Create a problem instance
mu = 1

# Add constraints on our variables
constraints = Constraint[
    position[:, i+1] == position[:, i] + h * velocity[:, i] for i in 1:T-1
]

for i in 1:T-1
    acceleration = force[:, i] / mass + g - drag * velocity[:, i]
    push!(constraints, velocity[:, i+1] == velocity[:, i] + h * acceleration)
end

# Add position constraints
push!(constraints, position[:, 1] == 0)
push!(constraints, position[:, T] == final_position)

# Add velocity constraints
push!(constraints, velocity[:, 1] == initial_velocity)
push!(constraints, velocity[:, T] == 0)

# Solve the problem
problem = minimize(sumsquares(force), constraints)
solve!(problem, SCS.Optimizer; silent = true)

Problem statistics
  problem is DCP         : true
  number of variables    : 3 (598 scalar elements)
  number of constraints  : 202 (404 scalar elements)
  number of coefficients : 605
  number of atoms        : 1_895

Solution summary
  termination status : OPTIMAL
  primal status      : FEASIBLE_POINT
  dual status        : FEASIBLE_POINT
  objective value    : 17849.7044

Expression graph
  minimize
   └─ qol (convex; positive)
      ├─ reshape (affine; real)
      │  └─ 2×99 real variable (id: 100…864)
      └─ [1;;]
  subject to
   ├─ == constraint (affine)
   │  └─ + (affine; real)
   │     ├─ index (affine; real)
   │     │  └─ …
   │     └─ Convex.NegateAtom (affine; real)
   │        └─ …
   ├─ == constraint (affine)
   │  └─ + (affine; real)
   │     ├─ index (affine; real)
   │     │  └─ …
   │     └─ Convex.NegateAtom (affine; real)
   │        └─ …
   ├─ == constraint (affine)
   │  └─ + (affine; real)
   │     ├─ index (affine; real)
   │     │  └─ …
   │     └─ Convex.NegateAtom (affine; real)
   │        └─ …
   ⋮

We can plot the trajectory taken by the object.

pos = evaluate(position)
plot([pos[1, 1]], [pos[2, 1]], st = :scatter, label = "initial point")
plot!([pos[1, T]], [pos[2, T]], st = :scatter, label = "final point")
plot!(pos[1, :], pos[2, :], label = "trajectory")

We can also see how the magnitude of the force changes over time.

plot(vec(sum(evaluate(force) .^ 2, dims = 1)), label = "force (magnitude)")

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