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Optimal control and stabilization of building maintenance units based on minimum principle

  • * Corresponding author: Shi'an Wang

    * Corresponding author: Shi'an Wang 
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  • In this paper we present a mathematical model describing the physical dynamics of a building maintenance unit (BMU) equipped with reaction jets. The momentum provided by reaction jets is considered as the control variable. We introduce an objective functional based on the deviation of the BMU from its equilibrium state due to external high-wind forces. Pontryagin minimum principle is then used to determine the optimal control policy so as to minimize possible deviation from the rest state thereby increasing the stability of the BMU and reducing the risk to the workers as well as the public. We present a series of numerical results corresponding to three different scenarios for the formulated optimal control problem. These results show that, under high-wind conditions the BMU can be stabilized and brought to its equilibrium state with appropriate controls in a short period of time. Therefore, it is believed that the dynamic model presented here would be potentially useful for stabilizing building maintenance units thereby reducing the risk to the workers and the general public.

    Mathematics Subject Classification: Primary: 49J15; Secondary: 93C95.

    Citation:

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  • Figure 1.  The schematic of a BMU

    Figure 2.  The schematic of the BMU body

    Figure 3.  Simulation results of scenario 1

    Figure 4.  Simulation results corresponding to $ U_{1} $ in scenario 2

    Figure 5.  Simulation results corresponding to $ U_{2} $ in scenario 2

    Figure 6.  Simulation results of case 1 in scenario 3

    Figure 7.  Simulation results of case 2 in scenario 3

    Table 1.  Definition of Notations

    Notation Description
    $ x_{1} = \omega_{x} $ First component of the angular velocity
    $ x_{2} = \omega_{y} $ Second component of the angular velocity
    $ x_{3} = \gamma_{x} $ First component of the unit vertical vector $ \gamma $
    $ x_{4} = \gamma_{y} $ Second component of the unit vertical vector $ \gamma $
    $ x_{5} = \gamma_{z} $ Third component of the unit vertical vector $ \gamma $
    $ \psi $ Costate vector (adjoint state)
    $ C $ Constant inertia matrix
    $ I=[0,T] $ Total operating period in seconds
    $ U $ Control (decision) constraint set
    $ {\mathcal U}_{ad} $ Set of admissible controls
    $ \underline{u} $ Lower bound of the control variable
    $ \overline{u} $ Upper bound of the control variable
    $ J(u) $ Objective (cost) functional
    $ \ell $ Integrand of running cost
    $ \Phi $ Terminal cost
    $ H $ Hamiltonian function
    $ V(x) $ Lyapunov function candidate
    $ x^{d} $ Desired state during the operating period
    $ \bar{x} $ Target state at the terminal time
    $ x^{e} $ Equilibrium state of the system
    $ \langle a,\; b \rangle $ Scalar product of vectors $ a $ and $ b $
    $ a \times b $ Cross product of vectors $ a $ and $ b $
    $ x^{T} $ Transpose of vector $ x $
     | Show Table
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    Algorithm 1: Computational Algorithm $ (\bullet) $
    Require:
    Choose the appropriate initial state $ x(0) $;
    Set the length of time horizon $ T \in R^{+} $ and the number of subintervals (of equal length) $ N \in \mathbb{Z}^{+} $;
    Set step size $ \epsilon $, stopping criterion $ \tau $, maximum number of iterations $ K $ and control bounds $ \underline{u}, \; \overline{u} $.
    Ensure:
    Optimal cost $ J^o $;
    Optimal state trajectory $ x^{o} $;
    Optimal control trajectory $ u^o $.

    1: Subdivide equally the time horizon $ I = [0, T] $ into $ N $ subintervals and assume the control function is piecewise-constant. That is, $ u^{n}(t) = u^{n}(t_{i}) $, for $ t \in [t_{i}, t_{i+1}),\; i = 0, 1, \cdots, N-1 $, where $ u^{n}(t),\; t \in I $ is the control (decision) policy at the $ n $th iteration (starting from $ n = 0 $).
    2: Integrate the state equations from 0 to $ T $ with initial state $ x(0) = x_{0} $ and the assumed controls $ u^{(n)} \equiv u^{n}(t),\; t \in I $, store the obtained state trajectory $ x^{(n)} $ and the control vector $ u^{(n)} $.
    3: Use $ x^{(n)} $ and $ u^{(n)} $ to integrate the adjoint equations backward in time starting from the costate $ \psi^{(n)}(T) $ at the terminal time. The terminal costate is given by $ \psi^{(n)}(T) = \Phi_{x}(x^{(n)}(T)) $ where $ \Phi $ is the terminal cost.
    4: Use the triple {$ u^{(n)},\; x^{(n)},\; \psi^{(n)} $} to compute the gradient $ g_{n}(t) = \frac{\partial H}{\partial u^{(n)}}(x^{(n)},\; u^{(n)},\; \psi^{(n)}) = H_{u}(x^{(n)},\; u^{(n)},\; \psi^{(n)}) $ and store this vector.
    5: Compute the cost functional $ J^{(n)}(u) $ using equation (19) and store this value.
    6: If $ \| g_{n} \| < \tau $ then set
    $ u^{o} = u^{(n)} $, $ J^{o} = J^{(n)} $
    return
    Otherwise, go to Step 7.
    7: Construct the control policy for the next iteration as
    $ u^{(n+1)}(t) = u^{(n)}(t) - \epsilon g_{n}(t),\; t \in I $ by choosing an appropriate $ \epsilon \in (0,1) $ such that $ u^{(n+1)} \in U $. For the chosen $ \epsilon $, if $ u^{(n+1)} > \overline{u} $ set $ u^{(n+1)} = \overline{u} $; if $ u^{(n+1)} < \underline{u} $ set $ u^{(n+1)} = \underline{u} $.
    8: If $ n < K $ then set
    $ n = n + 1 $, go to Step 2.
    Otherwise, display "Stopped before required residual is obtained''.
     | Show Table
    DownLoad: CSV
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