This study proposes a method for the static output feedback (SOF) stabilization of discrete time linear time invariant (LTI) systems by using a low number of sensors. The problem is investigated in two parts. First, the optimal sensor placement is formulated as a quadratic mixed integer problem that minimizes the required input energy to steer the output to a desired value. Then, the SOF stabilization, which is one of the most fundamental problems in the control research, is investigated. The SOF gain is calculated as a projected solution of the Hamilton-Jacobi-Bellman (HJB) equation for discrete time LTI system. The proposed method is compared with several examples from the literature.
BaraGIBoutayebM (2005) Static output feedback stabilization with performance for linear discrete-time systems. IEEE Transactions on Automatic Control50(2): 250–254.
2.
BelabbasMA (2016) Geometric methods for optimal sensor design. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences472(2185): 20150312.
3.
BelabbasMAChenX (2018) Sensor placement for optimal estimation of vector-valued diffusion processes. Systems & Control Letters121: 24–30.
4.
BenderD (1987) Lyapunov-like equations and reachability/observability gramians for descriptor systems. IEEE Transaction on Automatic Control32(4): 343–348.
5.
BertsekasDP (1995) Dynamic programming and optimal control. Vol. 1. Athena scientific optimization and computation series. Belmont, MA: Athena Scientific.
6.
CasadeiGCanudas-de-WitCZampieriS (2018) Controllability of large-scale networks: An output controllability approach. In: 57th IEEE Conference on Decision and Control, Miami, FL, USA, 17–19 December 2018, pp. 5886–5891. IEEE.
7.
ÇelaANiculescuSINatowiczRReamaA (2018) Controllability and observability gramians as information metrics for optimal design of networked control systems. Mechanical Engineering140(12): S8–S15.
8.
ChenXBelabbasMABaT (2019) Controlling and stabilizing a rigid formation using a few agents. SIAM Journal of Control Optimization57(1): 104–128.
9.
CrusiusCARTrofinoA (1999) Sufficient LMI conditions for output feedback control problems. IEEE Transactions on Automatic Control44(5): 1053–1057.
10.
Edelstein-KeshetL (2005) Mathematical Models in Biology. Vancouver, Canada: SIAM.
11.
FuM (2004) Pole placement via static output feedback is NP-hard. IEEE Transactions on Automatic Control49(5): 855–857.
12.
GadewadikarJLewisFL (2006) Aircraft flight controller tracking design using static output-feedback. Transactions of the Institute of Measurement and Control28(5): 429–440.
13.
GadewadikarJLewisFLXieL, et al. (2007) Parameterization of all stabilizing static state-feedback gains: Application to output-feedback design. Automatica43(9): 1597–1604.
14.
GamrathGAndersonDBestuzhevaK, et al. (2020) The SCIP Optimization Suite 7.0. Technical report, Optimization Online.
15.
GarciaGPradinBZengF (2001) Stabilization of discrete time linear systems by static output feedback. IEEE Transactions on Automatic Control46(12): 1954–1958.
16.
GarvieMR (2007) Finite-difference schemes for reaction–diffusion equations modeling predator-prey interactions in MATLAB. Bulletin of Mathematical Biology69(3): 931–956.
17.
GuG (1990) On the existence of linear optimal control with output feedback. SIAM Journal on Control and Optimization28(3): 711–719.
18.
HaleviYWagner-NachshoniC (2006) Transfer function modeling of multi-link flexible structures. Journal of Sound and Vibration296(1): 73–90.
19.
HiramotoKDokiHObinataG (2000) Optimal sensor/actuator placement for active vibration control using explicit solution of algebraic riccati equation. Journal of Sound and Vibration229(5): 1057–1075.
20.
KlicksteinIShirinASorrentinoF (2017) Energy scaling of targeted optimal control of complex networks. Nature Communications8.
LancasterPRodmanL (1995) Algebraic Riccati Equations. New York: Oxford University Press.
23.
LewisFLVrabieDSyrmosVL (1995) Optimal Control. 2nd edition. New York: Wiley.
24.
LiuYYSlotineJJBarabásiAL (2011) Controllability of complex networks. Nature473(7346): 167–173.
25.
MarxBKoenigDGeorgesD (2002) Optimal sensor/actuator location for descriptor systems using Lyapunov-like equations. In: Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, USA, 10–13 December 2002, volume 4. pp. 4541–4542. IEEE.
26.
MercadoALiuKR (2001) NP-hardness of the stable matrix in unit interval family problem in discrete time. Systems & Control Letters42(4): 261–265.
27.
MüllerFJSchuppertA (2011) Few inputs can reprogram biological networks. Nature478(7369): E4–E4.
28.
NemirovskiiA (1993) Several NP-hard problems arising in robust stability analysis. Mathematics of Control, Signals and Systems6(2): 99–105.
29.
Palacios-QuiñoneroFRubió-MassegúJRossellJKarimiH (2014) Feasibility issues in static output-feedback controller design with application to structural vibration control. Journal of the Franklin Institute351(1): 139–155.
30.
Palacios-QuiñoneroFRubió-MassegúJRossellJMKarimiHR (2012) Discrete-time static output-feedback semi-decentralized controller design: An application to structural vibration control. In: 2012 American Control Conference, Montreal, CA, 27–29 June 2012, pp. 6126–6131. IEEE.
31.
PolyakBTShcherbakovPS (2005) Hard problems in linear control theory: Possible approaches to solution. Automation and Remote Control66(5): 681–718.
32.
RosinováDVeselVKučeraV (2003) A necessary and sufficient condition for static output feedback stabilizability of linear discrete-time systems. Kybernetika39(4): 447–459.
33.
RughWJ (1996) Linear System Theory. Vol. 2. Upper Saddle River, NJ: Prentice Hall.
34.
SadabadiMSPeaucelleD (2016) From static output feedback to structured robust static output feedback: A survey. Annual Reviews in Control42(2016): 11–26.
35.
ShakerHRTahavoriM (2013) Optimal sensor and actuator location for unstable systems. Journal of Vibration and Control19(12): 1915–1920.
36.
SmithMSondergeldK (1986) On the order of stable compensators. Automatica22(1): 127–129.
37.
StevensBLLewisFLJohnsonEN (2016) Aircraft Control and Simulation: Dynamics, Controls Design, and Autonomous Systems. New Jersey: John Wiley & Sons.
38.
SummersTHCortesiFLLygerosJ (2016) On submodularity and controllability in complex dynamical networks. IEEE Transactions on Control of Network Systems3(1): 91–101.
39.
SummersTHLygerosJ (2014) Optimal sensor and actuator placement in complex dynamical networks. IFAC Proceedings Volumes47(3): 3784–3789.
40.
TokerOÖzbayH (1995) On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback. In: Proceedings of 1995 American Control Conference, Seattle, WA, USA, 21–23 June 1995, volume 4, pp. 2525–2526. IEEE.
41.
Trofino-NetoAKučeraV (1993) Stabilization via static output feedback. IEEE Transactions on Automatic Control38(5): 764–765.
42.
TuringAM (1990) The chemical basis of morphogenesis. Bulletin of Mathematical Biology52(1): 153–197.
43.
van de WalMde JagerB (2001) A review of methods for input/output selection. Automatica37(4): 487–510.
YücesoyVÖzbayH (2019) On the real, rational, bounded, unit interpolation problem in and its applications to strong stabilization. Transactions of the Institute of Measurement and Control41(2): 476–483.
46.
ZhangMMorrisK (2018) Sensor choice for minimum error variance estimation. IEEE Transactions on Automatic Control63(2): 315–330.
47.
ZhouKDoyleJGloverK (1996) Robust and Optimal Control. Volume 40. New Jersey: Prentice Hall.
48.
ZhouKSalomonGWuE (1999) Balanced realization and model reduction for unstable systems. International Journal of Robust and Nonlinear Control9(3): 183–198.
