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July  2021, 17(4): 1713-1727. doi: 10.3934/jimo.2020041

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

 School of EECS, University of Ottawa, 800 King Edward Ave. Ottawa, ON K1N 6N5, Canada

* Corresponding author: Shi'an Wang

Received  August 2019 Revised  October 2019 Published  July 2021 Early access  February 2020

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.

Citation: Shi'an Wang, N. U. Ahmed. Optimal control and stabilization of building maintenance units based on minimum principle. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1713-1727. doi: 10.3934/jimo.2020041
##### References:
 [1] N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/6262. [2] N. U. Ahmed, Elements of Finite Dimensional Systems and Control Theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, 37, John Wiley & Sons, Inc., New York, 1988. [3] T. Ahmed and N.U. Ahmed, Optimal Control of Antigen-Antibody Interactions for Cancer Immunotherapy, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, 26 (2019), 135-152. [4] D. Allen, What is building maintenance?, Facilities, 11 (1993), 7-12.  doi: 10.1108/EUM0000000002230. [5] Building Maintenance Units, Report of Alimak Group AB. Available from: https://alimakservice.com/building-maintenance-units/. [6] T. R. Chandrupatla, A. D. Belegundu and T. Ramesh, et al., Introduction to Finite Elements in Engineering, Prentice Hall, 2002. [7] J. C. P. Cheng, W. Chen and Y. Tan, et al., A BIM-based decision support system framework for predictive maintenance management of building facilities, 16th International Conference on Computing in Civil and Building Engineering, Osaka, Japan, 2016, 711–718. [8] T. Glad and L. Ljung, Control Theory, CRC Press, London, 2014.  doi: 10.1201/9781315274737. [9] R. M. W. Horner, M. A. El-Haram and A. K. Munns, Building maintenance strategy: A new management approach, J. Quality Maintenance Engineering, 3 (1997), 273-280.  doi: 10.1108/13552519710176881. [10] Instability of Building Maintenance Units, WorkSafe Victoria, 2018. Available from: https://www.worksafe.vic.gov.au/safety-alerts/instability-building-maintenance-units. [11] C. H. Ko, RFID-based building maintenance system, Automat. Construction, 18 (2009), 275-284.  doi: 10.1016/j.autcon.2008.09.001. [12] H. Lind and H. Muyingo, Building maintenance strategies: Planning under uncertainty, Property Management, 30 (2012), 14-28.  doi: 10.1108/02637471211198152. [13] P. Maryam, N.U. Ahmed and M.C.E. Yagoub, Optimum Decision Policy for Replacement of Conventional Energy Sources by Renewable Ones, International Journal of Energy Science, 3 (2013), 311-319.  doi: 10.14355/ijes.2013.0305.03. [14] I. Motawa and A. Almarshad, A knowledge-based BIM system for building maintenance, Automat. Construction, 29 (2013), 173-182.  doi: 10.1016/j.autcon.2012.09.008. [15] K. Ogata and Y. Yang, Modern Control Engineering, Prentice Hall, New Jersey, 2002. [16] L. S. Pontryagin, V. G. Boltyanskii and R. V. Gamkrelidze, et al., The Mathematical Theory of Optimal Processes, The Macmillan Co., New York, 1964. [17] I. H. Seeley, Building Maintenance, Building and Surveying Series, Palgrave, London, 1987. doi: 10.1007/978-1-349-18925-0. [18] J. Shen, A. K. Sanyal and N. A. Chaturvedi, et al., Dynamics and control of a 3D pendulum, 43$^rd$ IEEE Conference on Decision and Control, Atlantis, Bahamas, 2004, 323–328. doi: 10.1109/CDC.2004.1428650. [19] M.M. Suruz, N.U. Ahmed and M. Chowdhury, Optimum policy for integration of renewable energy sources into the power generation system, Energy Economics, 34 (2012), 558-567.  doi: 10.1016/j.eneco.2011.08.002. [20] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, 55, John Wiley & Sons, Inc., New York, 1991. doi: 20.500.11937/24319. [21] S. Wang and N.U. Ahmed, Dynamic model of urban traffic and optimum management of its flow and congestion, Dynamic Systems and Applications, 26 (2017), 575-588.  doi: 10.12732/dsa.v26i34.12. [22] S. Wang, N.U. Ahmed and T.H. Yeap, Optimum management of urban traffic flow based on a stochastic dynamic model, IEEE Transactions on Intelligent Transportation Systems, 20 (2019), 4377-4389.  doi: 10.1109/TITS.2018.2884463. [23] X. Wang, Solving optimal control problems with MATLAB: Indirect methods, ISE Dept., NCSU, Raleigh, NC, 2009. [24] D. V. Zenkov, On Hamel's equations, Theoret. Appl. Mechanics, 43 (2016), 191-220.  doi: 10.2298/TAM160612011Z. [25] D. V. Zenkov, M. Leok and A. M. Bloch, Hamel's formalism and variational integrators on a sphere, 51st IEEE Conference on Decision and Control, Hawaii, 2012, 7504–7510. doi: 10.1109/CDC.2012.6426779.

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##### References:
 [1] N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/6262. [2] N. U. Ahmed, Elements of Finite Dimensional Systems and Control Theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, 37, John Wiley & Sons, Inc., New York, 1988. [3] T. Ahmed and N.U. Ahmed, Optimal Control of Antigen-Antibody Interactions for Cancer Immunotherapy, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, 26 (2019), 135-152. [4] D. Allen, What is building maintenance?, Facilities, 11 (1993), 7-12.  doi: 10.1108/EUM0000000002230. [5] Building Maintenance Units, Report of Alimak Group AB. Available from: https://alimakservice.com/building-maintenance-units/. [6] T. R. Chandrupatla, A. D. Belegundu and T. Ramesh, et al., Introduction to Finite Elements in Engineering, Prentice Hall, 2002. [7] J. C. P. Cheng, W. Chen and Y. Tan, et al., A BIM-based decision support system framework for predictive maintenance management of building facilities, 16th International Conference on Computing in Civil and Building Engineering, Osaka, Japan, 2016, 711–718. [8] T. Glad and L. Ljung, Control Theory, CRC Press, London, 2014.  doi: 10.1201/9781315274737. [9] R. M. W. Horner, M. A. El-Haram and A. K. Munns, Building maintenance strategy: A new management approach, J. Quality Maintenance Engineering, 3 (1997), 273-280.  doi: 10.1108/13552519710176881. [10] Instability of Building Maintenance Units, WorkSafe Victoria, 2018. Available from: https://www.worksafe.vic.gov.au/safety-alerts/instability-building-maintenance-units. [11] C. H. Ko, RFID-based building maintenance system, Automat. Construction, 18 (2009), 275-284.  doi: 10.1016/j.autcon.2008.09.001. [12] H. Lind and H. Muyingo, Building maintenance strategies: Planning under uncertainty, Property Management, 30 (2012), 14-28.  doi: 10.1108/02637471211198152. [13] P. Maryam, N.U. Ahmed and M.C.E. Yagoub, Optimum Decision Policy for Replacement of Conventional Energy Sources by Renewable Ones, International Journal of Energy Science, 3 (2013), 311-319.  doi: 10.14355/ijes.2013.0305.03. [14] I. Motawa and A. Almarshad, A knowledge-based BIM system for building maintenance, Automat. Construction, 29 (2013), 173-182.  doi: 10.1016/j.autcon.2012.09.008. [15] K. Ogata and Y. Yang, Modern Control Engineering, Prentice Hall, New Jersey, 2002. [16] L. S. Pontryagin, V. G. Boltyanskii and R. V. Gamkrelidze, et al., The Mathematical Theory of Optimal Processes, The Macmillan Co., New York, 1964. [17] I. H. Seeley, Building Maintenance, Building and Surveying Series, Palgrave, London, 1987. doi: 10.1007/978-1-349-18925-0. [18] J. Shen, A. K. Sanyal and N. A. Chaturvedi, et al., Dynamics and control of a 3D pendulum, 43$^rd$ IEEE Conference on Decision and Control, Atlantis, Bahamas, 2004, 323–328. doi: 10.1109/CDC.2004.1428650. [19] M.M. Suruz, N.U. Ahmed and M. Chowdhury, Optimum policy for integration of renewable energy sources into the power generation system, Energy Economics, 34 (2012), 558-567.  doi: 10.1016/j.eneco.2011.08.002. [20] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, 55, John Wiley & Sons, Inc., New York, 1991. doi: 20.500.11937/24319. [21] S. Wang and N.U. Ahmed, Dynamic model of urban traffic and optimum management of its flow and congestion, Dynamic Systems and Applications, 26 (2017), 575-588.  doi: 10.12732/dsa.v26i34.12. [22] S. Wang, N.U. Ahmed and T.H. Yeap, Optimum management of urban traffic flow based on a stochastic dynamic model, IEEE Transactions on Intelligent Transportation Systems, 20 (2019), 4377-4389.  doi: 10.1109/TITS.2018.2884463. [23] X. Wang, Solving optimal control problems with MATLAB: Indirect methods, ISE Dept., NCSU, Raleigh, NC, 2009. [24] D. V. Zenkov, On Hamel's equations, Theoret. Appl. Mechanics, 43 (2016), 191-220.  doi: 10.2298/TAM160612011Z. [25] D. V. Zenkov, M. Leok and A. M. Bloch, Hamel's formalism and variational integrators on a sphere, 51st IEEE Conference on Decision and Control, Hawaii, 2012, 7504–7510. doi: 10.1109/CDC.2012.6426779.
The schematic of a BMU
The schematic of the BMU body
Simulation results of scenario 1
Simulation results corresponding to $U_{1}$ in scenario 2
Simulation results corresponding to $U_{2}$ in scenario 2
Simulation results of case 1 in scenario 3
Simulation results of case 2 in scenario 3
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 $x$
 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 $x$
 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''.
 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''.
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