State-feedback control of a reaction-wheel inverted pendulum prototype

1 Introduction

The inverted pendulum is a popular benchmark in nonlinear control theory (Boubaker, 2013; Boubaker and Iriarte, 2017). Although the system is conceptually simple, its open-loop instability in the upright position and inherent nonlinearity have kept it at the center of teaching and applied research for decades, with a literature that ranges from conventional approaches such as proportional–integral–derivative (PID) and fuzzy-logic control (Silva et al., 2013), sliding-mode and back-stepping control (Irfan et al., 2024), PID, pole placement, and the Linear Quadratic Regulator (LQR) (Baltazar et al., 2024), and developments based on H ∞ synthesis (Alves et al., 2022) and discrete-time H 2 control (Varaschim et al., 2024). Among the available variants, the Reaction-Wheel Inverted Pendulum (RWIP), introduced as a research and teaching benchmark by Spong et al. (2001) and consolidated as a didactic platform by Block et al. (2007), is one of the simplest mechanical systems that exhibits all the difficulties associated with the stabilization of an unstable equilibrium while remaining tractable for a wide range of control synthesis methods.

The RWIP is a planar two-degree-of-freedom mechanical system: the pendulum angle σ and the wheel angle ϕ are independent generalized coordinates. Only one control input is available, the voltage applied to the direct-current (DC) motor that drives the reaction wheel; the RWIP is therefore underactuated in the standard sense (Spong et al., 2001; Fantoni and Lozano, 2002). The wheel coordinate ϕ is, however, cyclic: only its derivative ϕ ̇ enters the equations of motion, which permits a reduction of the state from four to three variables, x = [ σ , σ ̇ , ϕ ̇ ] T, without loss of information relevant for stabilization. This three-state-variable form is adopted here, in agreement with the formulation of Spong et al. (2001) and with subsequent works on the same prototype (Santos et al., 2026).

Reaction wheels are momentum-exchange devices, and their use extends well beyond the laboratory. Constrained spacecraft attitude control and reaction-wheel desaturation (Castroviejo-Fernandez and Kolmanovsky, 2023) rely on the same physical principle that governs the RWIP, which underscores the relevance of comparing different stabilization strategies on a benchmark that captures the essential dynamics. Recent variants of the same principle include the One-Wheel Cubli (Hofer et al., 2023), a 3D pendulum stabilized by a single reaction wheel, and the inverted pendulum with two reaction wheels proposed by Trentin et al. (2020b), which illustrate the continued vitality of the topic.

A practical difficulty in the RWIP is that, starting from the stable downward equilibrium, the actuator alone usually cannot drive the pendulum to the upright position; the limited DC-motor torque must be combined with a maneuver that injects energy through the system’s natural oscillations. Åström and Furuta (2000) formalized this idea for accelerated-pivot pendulums by means of an energy-control law derived from a Lyapunov function, which admits a simplified bang-bang implementation that drives the input to its saturation limits according to the sign of the pendulum velocity. Spong et al. (2001), in turn, proposed a passivity-based swing-up for the RWIP, built upon partial feedback linearization and a continuous control law whose convergence is established through the invariance principle; further developments along this line were presented by Fantoni et al. (2001). The bang-bang variant of the energy-control law has been adopted in subsequent low-cost RWIP implementations (da Silva et al., 2026), given its natural suitability for embedded prototypes; for instance, experimental switching pipelines that combine energy-based swing-up with linear stabilization on the RWIP have been reported by Srinivas and Behera (2008). Once the pendulum reaches a neighborhood of the upright position, a stabilization controller must take over; a switching strategy between swing-up and balance is, therefore, a structural feature of any complete RWIP control scheme (Spong et al., 2001).

Whenever the linearized model is controllable and the full state is available for feedback, a wide range of stabilization techniques is applicable, including robust H ∞/H 2 and sliding-mode, as the works discussed above illustrate. Among them, pole placement and the LQR were selected in this work because they are the most widely used full-state-feedback baselines, rely on a small and interpretable set of design parameters, and are directly comparable on the same model structure, which suits the didactic purpose of the platform (Ogata, 2010). Pole placement assigns the closed-loop eigenvalues directly, offering a transparent design parameter at the cost of providing no explicit handle on control effort. LQR, in contrast, returns the gain that minimizes a quadratic performance index penalizing both state deviation and control effort, but only indirectly shapes the closed-loop dynamics. Both techniques have been thoroughly studied in isolation: pole placement has been applied, among others, to lateral control of autonomous ground vehicles (Rahim et al., 2025), to a laboratory air-levitation system (Rosinova et al., 2024), and to tracking control of a car-and-beam prototype (Camargo et al., 2024); LQR has been used on the cart-and-pole pendulum (Nasir et al., 2008), on nautical stabilization (Passos et al., 2018), and on an Aeropendulum benchmark (Yamanaka et al., 2022).

For the RWIP itself, the literature on direct comparison between pole placement and LQR is more fragmented. Most works either compare the two techniques solely in simulation (Asavakanoksilp and Chancharoen, 2025; Li et al., 2024; Rani and Kamlu, 2021; Sweiti and Tamimi, 2023), or validate only a single technique on hardware. Bapiraju et al. (2004) apply pole placement on three linearized models of the RWIP and validate them in real time on a mechatronics control kit, but without considering LQR synthesis. Karhan et al. (2024) present a tutorial-style implementation of LQR alone on an STM32F4 target, reporting a 20° recovery angle on a reaction-wheel pendulum. Trentin et al. (2020a) implement PID controllers on classical and dual-wheel reaction-wheel pendulums using low-cost hardware, comparing the two configurations but without considering state-feedback synthesis methods. Zaborniak et al. (2024) report a comparative study of cascade PID and an LQR coupled with a Kalman filter on a low-cost wheel-based inverted pendulum implemented on an STM32 board, with disturbance-rejection tests but without pole placement. The closest prior work is Michalski et al. (2025), who recently compared pole placement and LQR across three state-space representations of a reaction-wheel pendulum, with experimental validation at both equilibria; that study varies the model representation while holding the synthesis methods fixed, whereas the present work fixes the representation and embeds the comparison in a complete pipeline that includes a bang-bang energy-injection swing-up and a switching law. The two studies are therefore complementary.

On the same prototype used here, a linear-matrix-inequality (LMI)-based state feedback with prescribed decay rate and gain-norm constraints was previously applied, for stabilization only and without swing-up (Santos et al., 2026); an independent study with a similar prototype, also based on an ESP32 microcontroller, has focused on PID, eigenvalue-placement, and sliding-mode control combined with a switching swing-up (da Silva et al., 2026).

Within this landscape, the present paper offers a unified experimental comparison of pole placement and LQR for the stabilization of a single low-cost RWIP prototype, integrated with a bang-bang energy-injection swing-up and a switching law, and evaluated through the integral of the absolute error (IAE), the integral of the squared error (ISE), and their time-weighted counterparts (ITAE and ITSE) under the same disturbance scenario. To the best of our knowledge, no previous work reports a same-prototype, same-disturbance-scenario experimental comparison of pole placement and LQR embedded in a complete swing-up/switching/stabilization pipeline and quantified through the four integral indices IAE, ISE, ITAE, and ITSE; this unified experimental comparison, carried out on a low-cost and easily replicable platform, is the main contribution of this work. The contributions are as follows:

• A description of an RWIP prototype built at the Industrial Automation and Control Laboratory of IFPR, Campus Jacarezinho, which is instrumented with an ESP32 microcontroller and integrated with MATLAB/Simulink for real-time control and data acquisition with a sampling period of 1 ms;
• The design of two full-state feedback stabilization controllers (pole placement and LQR) on a state-space model that couples the Lagrangian dynamics of the pendulum with the DC-motor dynamics, linearized around the unstable upright equilibrium, and their integration into a complete control pipeline in which a simplified bang-bang energy-injection swing-up, following Åström and Furuta (2000), raises the pendulum and an angle-based switching law hands control over to the stabilizing law;
• An experimental comparison of the two stabilization controllers on the prototype, including disturbance-rejection tests, with performance assessed through time-domain response and through the integral indices.

The remainder of the paper is organized as follows. Section 2 presents the RWIP dynamic model and the laboratory prototype. Section 3 describes the bang-bang swing-up law and the switching strategy between swing-up and stabilization. Section 4 details the design of the pole-placement and LQR stabilization controllers. Section 5 reports simulation and experimental results, and the comparison through the integral performance indices. Final remarks are presented in Section 6.

2 Reaction-wheel inverted pendulum: Modeling and prototype

2.1 Dynamic model

A schematic diagram of the RWIP is shown in Figure 1. The pendulum is a rigid rod that rotates freely about a horizontal pivot. A reaction wheel is mounted at one end of the rod and is driven by a brushed DC motor; a counterweight is fixed at the opposite end of the rod in order to reduce the gravitational torque about the pivot and, with it, the control effort required for stabilization. The angular position of the rod is denoted by σ ( t ), while the angular position of the reaction wheel is denoted by ϕ ( t ). The angle σ ( t ) is measured so that σ = 0 corresponds to the unstable upright equilibrium of the pendulum, in agreement with the convention adopted by da Silva et al. (2026) on a closely related prototype.

The geometry is summarized by the following variables: M a, M m, and M r are, respectively, the masses of the wheel-to-motor coupling, of the motor, and of the wheel, with M t r = M a + M m + M r representing the lumped wheel and motor assembly; M h 1 and M h 2 are the masses of the two halves of the rod, on the wheel side and on the counterweight side, respectively; M c is the counterweight mass; L h 1 and L h 2 are the distances from the pivot to the rod ends on each side, with L ch1 and L ch2 the distances from the pivot to the corresponding rod center of mass; L h 3 and L h 4 are the distances from the pivot to the centers of mass of M t r and M c; and I r and I h are the inertias of the wheel and of the rod about their own centers of mass, respectively.

For compactness, the auxiliary parameters used in the equations of motion follow da Silva et al. (2026):

M t r = M a + M m + M r , I p = 1 3 M h 1 L h 1 2 + 1 3 M h 2 L h 2 2 + M t r L h 3 2 + M c L h 4 2 + I h , M ̄ 1 = M h 1 L ch1 + M t r L h 3 , M ̄ 2 = M h 2 L ch2 + M c L h 4 . (1)

In Equation 1, where I p is the total moment of inertia of the rod and wheel assembly about the pivot, M ̄ 1 and M ̄ 2 are the equivalent gravitational moments on the wheel and counterweight sides, respectively, and I h accounts for the rod’s own inertia contribution to I p.

The model is derived from Lagrange’s formulation. Lagrange’s formulation was adopted for three reasons: (i) it yields the equations of motion of the two-degree-of-freedom underactuated system directly in terms of the generalized coordinates ( σ , ϕ ), without requiring the computation and elimination of the internal reaction forces at the pivot and at the motor mounting, as a Newton–Euler derivation would; (ii) the kinetic and potential energy functions constructed in the derivation are precisely the quantities manipulated by the energy-based swing-up law of Section 3, which gives methodological consistency between the modeling and the control design; and (iii) it is the standard route in the RWIP literature (Fantoni et al., 2001; Bapiraju et al., 2004; Block et al., 2007; Karhan et al., 2024), which facilitates comparison with previous works. The resulting equations of motion are, of course, equivalent to those obtained by any other correct formulation. The kinetic energy, given in Equation 2, is the sum of the contributions of the rod and the reaction wheel:

E c t = 1 2 I p σ ̇ 2 t + I r ϕ ̇ t σ ̇ t + 1 2 I r ϕ ̇ 2 t . (2)

Neglecting any elasticity in the motor shaft and in the rod, the potential energy reduces to the gravitational contribution in Equation 3:

E p g = M ̄ 1 g cos ⁡ σ t − 1 + M ̄ 2 g 1 − cos ⁡ σ t . (3)

The Lagrangian is L a g = L a g ( q i , q ̇ i , t ) = E c t − E p g, which is given in Equation 4 as,

L a g = 1 2 I p σ ̇ 2 t + I r ϕ ̇ t σ ̇ t + 1 2 I r ϕ ̇ 2 t + M ̄ 1 g 1 − cos ⁡ σ t + M ̄ 2 g cos ⁡ σ t − 1 . (4)

Application of the Euler-Lagrange equation (Equation 5)with the convention that the motor exerts a torque τ ( t ) on the rotor and an equal and opposite reaction torque on the pendulum (Spong et al., 2001; da Silva et al., 2026), yields the equations of motion of the RWIP:

d d t ∂ L a g ∂ q ̇ i − ∂ L a g ∂ q i = τ i , (5)

I p σ ̈ t + I r ϕ ̈ t − M ̄ 1 g ⁡ sin ⁡ σ t + M ̄ 2 g ⁡ sin ⁡ σ t = 0 , I r σ ̈ t + I r ϕ ̈ t = τ t . (6)

The presence of τ ( t ) in the second row alone reflects the chosen sign convention; eliminating ϕ ̈ between the two equations in Equation 6 recovers the form (Equation 7) below, in which − τ ( t ) appears explicitly as the torque acting on the pendulum.

Solving (Equation 6) for σ ̈ and ϕ ̈ produces the nonlinear modelin which C i = I p I r − I r 2 is a shorthand for the inertia constant.

σ ̈ t = 1 C i I r M ̄ 1 g ⁡ sin ⁡ σ t − I r M ̄ 2 g ⁡ sin ⁡ σ t − I r τ t , ϕ ̈ t = 1 C i I r M ̄ 2 g ⁡ sin ⁡ σ t − I r M ̄ 1 g ⁡ sin ⁡ σ t + I p τ t , (7)

The DC-motor dynamics are coupled to the mechanical model through the generated torque. Following the standard derivation for a permanent-magnet DC motor with the armature inductance neglected (da Silva et al., 2026; Ogata, 2010), the torque on the motor shaft isin which η is the gear ratio, K t is the torque constant, K v is the back-electromotive-force (back-EMF) constant, A m is the brush-friction coefficient, V ( t ) is the armature voltage, and R a is the armature resistance. For a brushed DC motor expressed in coherent SI units, K t and K v are numerically identical, as reported in Table 1. The model in Equation 8 includes the motor brush-friction term A m ϕ ̇ ( t ) but neglects viscous and Coulomb friction at the wheel bearing. Recent experimental work by da Silva et al. (2026) on a related RWIP prototype identifies these terms from free-oscillation tests as F v = 0.0012 Nm / ( rad / s ) and F c = 4.1 × 1 0 − 4 Nm, and reports that their omission contributes to imperfections in the mathematical model of the system. Their identification was not performed for the prototype considered here; the implications of this simplification are revisited in the discussion of the experimental results.

τ t = η K t R a V t − η 2 K t K v R a ϕ ̇ t − A m ϕ ̇ t , (8)

TABLE 1. DC-motor parameters (model CHP-36GP-555-ABHL), reproduced from Santos et al. (2026).

Substitution of (Equation 8) into (Equation 7) and linearization about the unstable upright equilibrium, where sin ⁡ σ ( t ) ≈ σ ( t ) for small σ ( t ), give the linearized RWIP model

σ ̈ t = I r M ̄ 1 g C i σ t − I r M ̄ 2 g C i σ t − I r η K t C i R a V t + I r η 2 K t K v C i R a ϕ ̇ t + I r A m C i ϕ ̇ t , ϕ ̈ t = I r M ̄ 2 g C i σ t − I r M ̄ 1 g C i σ t + I p η K t C i R a V t − I p η 2 K t K v C i R a ϕ ̇ t − I p A m C i ϕ ̇ t . (9)

Following the formulation of Spong et al. (2001), since ϕ ( t ) is a cyclic coordinate that does not enter the equations of motion, only ϕ ̇ ( t ) is required for control. The state vector is therefore reduced towith control input u ( t ) = V ( t ), and the state-space formwith state and input matrices

x t = σ t σ ̇ t ϕ ̇ t T , (10)

x ̇ t = A x t + B u t , (11)

A = 0 1 0 I r M ̄ 1 − M ̄ 2 g C i 0 I r η 2 K t K v C i R a + I r A m C i I r M ̄ 2 − M ̄ 1 g C i 0 − I p η 2 K t K v C i R a − I p A m C i , B = 0 − I r η K t C i R a I p η K t C i R a . (12)

The reduced three-state representation in Equation 10 follows the canonical formulation of the reaction-wheel pendulum introduced by Spong et al. (2001) and is the same adopted in da Silva et al. (2026); Santos et al. (2026).

2.2 Prototype

The laboratory prototype is shown in Figure 2 and follows the construction described in Santos et al. (2026). The reaction wheel ① is rigidly fixed to the shaft of a brushed DC motor ② (model CHP-36GP-555-ABHL, 12 V nominal); the torque produced by this assembly is the only actuation input of the system and acts directly on the pendulum body. The same DC-motor model is used in the independent low-cost RWIP of da Silva et al. (2026), supporting its suitability for embedded RWIP platforms. The angular position of the motor shaft, used to reconstruct ϕ ̇ ( t ), is read by a 350-pulse-per-revolution (PPR) incremental encoder coupled to the motor ③. A metallic rod ④ connects the wheel–motor assembly on one side to the counterweight ⑤ on the other. A slip ring with a central bore ⑥ routes the motor power leads through the pivot, allowing the rod to rotate freely without twisting the cables. The angular position of the pendulum, σ ( t ), is measured by a high-resolution incremental encoder LPD3806-600BM-G5-24C (600 PPR) ⑦. Power switching is performed by an HW-039 H-bridge ⑧ (43 A continuous, 5–36 V), which receives pulse-width-modulation (PWM) signals from an ESP32 microcontroller ⑨; the same microcontroller acquires encoder readings. The prototype is powered by a programmable bench supply, MPC-3003 ⑩, and equipped with an emergency stop button ⑪. The control algorithms run in MATLAB/Simulink on a desktop computer ⑫ (AMD Phenom™ II X2 B57 3.20 GHz, 12 GB RAM) that communicates with the ESP32 over a serial link, with a sampling period of T s = 1 ms. The angular velocities σ ̇ ( t ) and ϕ ̇ ( t ) required for state feedback are not measured directly; they are obtained by derivative filters applied to the encoder readings. The derivative filter used is described by the transfer function F ( s ) = 20 s s + 20, where s is the Laplace variable. The sampling period T s = 1 ms was selected so that the discrete-time implementation behaves as quasi-continuous with respect to the designed dynamics: the fastest closed-loop eigenvalue has magnitude below 10 rad/s (≈ 1.6 Hz), so the 1 kHz rate exceeds the closed-loop bandwidth by more than two orders of magnitude. This renders the continuous-time designs of Section 4 valid without a discrete-time redesign, preserves the response of the derivative filter (pole at 20 rad/s), and is within the acquisition and communication capabilities of the ESP32 and of the serial link. The same rate was used in previous studies on this platform (Santos et al., 2026).

The physical parameters of the prototype and of the DC motor are given in Tables 1, 2. The mechanical parameters were taken from the prior characterization of the same apparatus reported in Santos et al. (2026), which in turn relies on values originally identified by the UNESP–Ilha Solteira group; the same values appear in da Silva et al. (2026). The DC-motor parameters are nominal datasheet values; experimental identification was not performed.

TABLE 2. Physical parameters of the RWIP, reproduced from Santos et al. (2026).

3 Swing-up and switching control

A practical difficulty of the RWIP, already pointed out in Section 1, is that the torque produced by the DC motor through the reaction wheel is not sufficient to lift the pendulum from the stable downward equilibrium to the unstable upright position by direct action of the stabilization controller (da Silva et al., 2026). The pendulum must therefore be brought close to the upright position by a separate maneuver before the stabilization controller is engaged. This requirement is specific to low-torque actuators: with sufficiently powerful motors, the pendulum can be driven to the upright region without a dedicated swing-up law (Trentin et al., 2020a).

Among the swing-up strategies discussed in the literature, this work adopts the energy-based approach of Åström and Furuta (2000), in the simplified bang-bang form. The bang-bang law commands the actuator to its saturation limits based on the sign of the pendulum velocity, injecting energy into the system at each oscillation cycle and progressively increasing the amplitude until the pendulum reaches a neighborhood of the upright position.

Throughout this section, the pendulum angle σ ( t ) follows the convention introduced in Section 2: σ = 0 corresponds to the unstable upright equilibrium and | σ | = π corresponds to the stable downward equilibrium. The swing-up maneuver, therefore, takes the pendulum from the downward equilibrium towards the upright region σ ( t ) ≈ 0, and the bang-bang law remains active while | σ ( t ) | ≥ π / 4.

The bang-bang law requires a non-zero angular velocity to be effective: at rest, with σ ̇ ( t ) = 0, its output is undefined. Since the maneuver starts from rest at the downward equilibrium (| σ ( t ) | = π, σ ̇ ( t ) = 0), an initialization pulse is added during the first second to displace the pendulum and produce a non-zero velocity; after this pulse, the bang-bang law alone drives the swing-up. The swing-up control input is therefore defined aswhere U = 8V is the bang-bang amplitude, set high enough to bring the pendulum into the neighborhood of the upright position within a few oscillation cycles.

u s w t = 1 + U ⋅ sgn σ ̇ t , 0 ≤ t ≤ 1 s , U ⋅ sgn σ ̇ t , t > 1 s , (13)

Once the pendulum approaches the upright region, the swing-up law must be replaced by the stabilization controller, since (Equation 13) is not designed to regulate the system about an equilibrium. A switching law is used for this purpose, partitioning the angular range of σ ( t ) into three operating regions.

• swing-up region, | σ ( t ) | ≥ π / 4: The pendulum is far from the upright position, and the law (Equation 13) is active, injecting energy into the system;
• coasting region, π / 12 < | σ ( t ) | < π / 4: The control input is set to zero, allowing the pendulum to enter the stabilization region through its own residual motion and smoothing the transition between the two controllers;
• stabilization region, | σ ( t ) | ≤ π / 12 (equivalently, within ±15° of the upright equilibrium): the state-feedback controller u ( t ) = − K x ( t ) is active.

The complete control law is thereforewith u s w ( t ) given by Equation 13 and x ( t ) the state vector defined in Equation 10. The state-feedback controller designed in Section 4 regulates σ ( t ) about the upright equilibrium σ ref = 0, with the closed-loop dynamics linearized as in Equation 9.

u t = u s w t , | σ t | ≥ π / 4 , 0 , π / 12 < | σ t | < π / 4 , − K x t , | σ t | ≤ π / 12 , (14)

Figure 3 illustrates the three operating regions of the control law (Equation 14), each associated with a different control action and identified by color. The green area corresponds to the swing-up region, where the bang-bang law u s w ( t ) injects energy into the system; the white region corresponds to the coasting region, where the input is set to zero and the pendulum evolves under its residual motion; the blue cone around the vertical corresponds to the stabilization region, where the state-feedback law u ( t ) = − K x ( t ) regulates σ ( t ) around the upright equilibrium. The boundaries between regions are placed at | σ ( t ) | = π / 4 and | σ ( t ) | = π / 12, in agreement with the thresholds adopted in Equation 14.

The size of the stabilization region, ±15° was set on the basis of bench tests, following the same principle adopted in similar low-cost implementations (da Silva et al., 2026): the threshold must be wide enough to capture the pendulum on its way up but narrow enough that the linearized model on which the state-feedback controller is designed remains a good approximation of the nonlinear dynamics. In addition, the threshold must keep the state-feedback demand within the actuator range: at | σ | = π / 12 rad, the position term of the state-feedback law alone already demands about 10.3 V of the available 12 V (see Section 5.1), which shows that substantially wider thresholds would not be implementable with the present motor. The adopted value is also consistent with related implementations: Karhan et al. (2024), for instance, report a 20 ° recovery angle on a comparable reaction-wheel pendulum. A simulation-based sensitivity analysis of this threshold is presented in Section 5.3.

4 Pole-placement and LQR control design

The state-space model of the RWIP, derived in Section 2, which is linearized around the unstable upright equilibrium and described by Equations 11 and 12, provides the basis for designing two state-feedback stabilization controllers. Both controllers aim to find the gain K ∈ R 1 × 3 that, when applied through the control law in Equation 15:yields the closed-loop dynamics in Equation 16:and makes the origin x ( t ) = 0, which corresponds to the upright equilibrium, asymptotically stable. The two design approaches differ in the criterion used to select K: pole placement assigns the eigenvalues of A − B K directly, while the LQR minimizes a quadratic performance index.

u t = − K x t , (15)

x ̇ t = A − B K x t , (16)

4.1 Pole placement

For any pair ( A , B ), the eigenvalues of A − B K can be assigned arbitrarily by choosing K appropriately if and only if ( A , B ) is controllable (Chen, 2013). The closed-loop solution of Equation 16 is

x t = e A − B K t x 0 . (17)

From Equation 17, so asymptotic stability of the origin (that is, x ( t ) → 0 as t → ∞ for any initial state x ( 0 )) is equivalent to all eigenvalues of A − B K having negative real part.

Controllability of the linearized RWIP model is verified by checking that the controllability matrix in Equation 18:has full row rank, with A and B in Equation 12. Given a target set of closed-loop eigenvalues { λ 1 , λ 2 , λ 3 } with negative real parts, the gain K that places the eigenvalues of A − B K at those locations is computed through the algorithm of Kautsky et al. (1985), which is numerically robust and serves as the foundation for standard control design software.

C = B A B A 2 B (18)

4.2 Linear Quadratic Regulator

The LQR approach selects K as the gain that minimizes the quadratic performance index

J = ∫ 0 ∞ x T t Q x t + u T t R u t d t . (19)

In Equation 19, where Q ∈ R n × n is symmetric positive semidefinite and R ∈ R m × m is symmetric positive definite. The matrix Q weights deviations of the state from the equilibrium, while R weights the control effort; their relative magnitudes therefore reflect the designer’s trade-off between regulation performance and actuator activity (Ogata, 2010).

The minimization of J subject to the system dynamics (Equation 11) yields the optimal gainwhere P = P T ≻ 0 is the unique stabilizing solution of the continuous-time algebraic Riccati equation

K = R − 1 B T P , (20)

A T P + P A − P B R − 1 B T P + Q = 0 . (21)

Existence and uniqueness of this stabilizing solution are guaranteed when ( A , B ) is stabilizable, R is positive definite, and ( A , Q 1 / 2 ) is detectable, where Q 1 / 2 denotes any matrix square root of Q (Lewis et al., 2012). Under these conditions, the closed-loop matrix A − B K obtained with the gain (Equation 20) is Hurwitz, and the state regulator (Equation 15) is asymptotically stable.

In practice, the choice of Q and R is guided by the desired balance between state deviation and control effort. A common starting point is the rule of thumb attributed to Bryson and Ho (1975). In this approach, the diagonal entries of Q and R are scaled by the inverses of the squared maximum admissible deviations of the corresponding state and input variables. This choice is then refined by iterative simulation.

5 Simulation and experimental results

5.1 Numerical model and controller design

Evaluating the state-space matrices (Equation 12) with the physical and DC-motor parameters of Tables 1 and 2 yields the numerical realization of the linearized RWIP model used for controller design:

A = 0 1.0000 0 1.8368 0 0.0932 − 1.8368 0 − 5.9504 , B = 0 − 0.7840 50.0627 . (22)

The controllability matrix C in Equation 18 formed from (Equation 22) has full rank ( det C = det [ B A B A 2 B ] = 55.6324 ≠ 0 ), so the linearized model is controllable and both state-feedback designs are applicable.

For the pole-placement controller, the closed-loop eigenvalues were chosen as

λ 1 = − 1 , λ 2 = − 2 , λ 3 = − 9 . (23)

Applying the pole-placement algorithm of Section 4.1 with the eigenvalues (Equation 23) yields the gain vector

K pp = − 39.3336 − 28.0054 − 0.3177 . (24)

For the LQR controller, the weighting matrices were selected as

Q = 0.2 0 0 0 0.2 0 0 0 0.025 , R = 1 . (25)

Solving the algebraic Riccati equation (Equation 21) with the weights (Equation 25) and substituting the result into the optimal-gain expression (Equation 20) yields

K lqr = − 38.6295 − 28.7031 − 0.3167 . (26)

The eigenvalues (Equation 23) were selected considering the actuator and instrumentation constraints: the dominant pair { − 1 , − 2 } rad/s yields a settling time of about 4 s after capture and, at the switching threshold | σ | = π / 12 rad, at which the state-feedback law takes over from the swing-up law (after the coasting phase), the position term of the resulting feedback alone demands | u | ≈ 10.3 V of the available 12 V, so that faster dominant dynamics would saturate the motor; the third eigenvalue, − 9 rad/s, mainly associated with the wheel velocity, is about 1 decade faster than the dominant pair and remains below the 20 rad/s bandwidth of the derivative filter. The LQR weights (Equation 25) were refined iteratively from a Bryson-type starting point until capture occurred without saturation and the closed-loop bandwidth was comparable to that of the pole-placement design; the closed-loop eigenvalues obtained with the gain (Equation 26) are approximately { − 1.34 , − 1.34 , − 9.9 } rad/s. The two controllers therefore operate at matched bandwidth, and the comparison isolates the effect of the synthesis criterion (direct eigenvalue assignment versus quadratic optimality) rather than the effect of arbitrarily different tunings; this also explains the similarity between the gains (Equation 24) and (Equation 26).

5.2 Performance indices

The controllers are compared using integral error indices, which provide a quantitative measure of the controlled variable’s overall response over time (Ogata, 2010). Four indices are used. The integral of the squared error (ISE), given in Equation 27,penalizes large errors strongly and is therefore sensitive to oscillations and to the speed of the initial response. The integral of the absolute error (IAE), defined in Equation 28,measures the accumulated error with less emphasis on the individual peaks than the ISE. The integral of the time-weighted absolute error (ITAE), expressed in Equation 29,weights errors more heavily as time progresses, and is therefore sensitive to long-lasting steady-state error. The integral of the time-weighted squared error (ITSE), given in Equation 30,combines the characteristics of the ISE and the ITAE, amplifying the penalty for large errors that occur later in the response. In all four indices, e ( t ) denotes the error of the controlled variable, the pendulum angle σ ( t ), relative to the upright reference. Lower index values correspond to better performance (Ogata, 2010).

I S E = ∫ 0 ∞ e 2 t d t , (27)

I A E = ∫ 0 ∞ | e t | d t , (28)

I T A E = ∫ 0 ∞ t | e t | d t , (29)

I T S E = ∫ 0 ∞ t e 2 t d t , (30)

5.3 Results and discussion

The gain vectors (Equation 24) and (Equation 26) were implemented on the physical prototype and in simulation. In both cases, the system was subjected to an external disturbance given byapplied to the motor voltage signal, with a control sampling period of 1 ms and a PWM carrier frequency of 490 Hz applied to the H-bridge. The control action was limited to ±12 V by a saturation block, protecting the DC motor. The simulation used the RWIP nonlinear model given in Equation 7, coupled with the motor model (Equation 8); the linearized model (Equation 9) was used only for controller design.

d t = 3 V if 25 ≤ t < 27 or 44 ≤ t < 46 , − 3 V if 27 ≤ t < 29 or 46 ≤ t < 48 , 0 V otherwise , (31)

The behavior of the RWIP in simulation and in the prototype experiment is shown in Figures 4, 5, respectively. A video recording of the experimental runs reported in this section is available at https://youtu.be/tKE4tn0lf1A.

In the simulation, the swing-up maneuver takes longer to raise the pendulum to the operating point than in the prototype experiment. This difference is attributed to physical effects not represented in the mathematical model, such as friction and mechanical backlash. In the absence of these effects, the simulated pendulum reaches a higher angular position than the one observed experimentally, which results in a longer settling time.

Two operating intervals are considered for the quantitative comparison. Table 3 reports the indices computed over the full response (integrating from 0 to 70 s), including the swing-up maneuver, while Table 4 reports the indices computed over the stabilization phase alone (integrating from 13 to 70 s), excluding the swing-up maneuver.

TABLE 3. Performance indices including the swing-up maneuver (0–70 s).

TABLE 4. Performance indices for the stabilization phase, excluding the swing-up maneuver (13–70 s).

Three pairwise comparisons can be drawn from Tables 3 and 4; the relative differences reported below are computed with respect to the larger value of each pair. (i) Simulation versus simulation: the two controllers are practically indistinguishable, with index differences below 0.5 % over the full response and below 2.6 % over the stabilization phase, which is consistent with the matched closed-loop bandwidth discussed in Section 5.1. (ii) Experiment versus experiment: over the full response, the differences remain below 5 %, with the LQR yielding marginally lower IAE, ISE, and ITSE and pole placement the lower ITAE, so neither controller is uniformly superior. Over the stabilization phase, the LQR attained IAE 16 % lower, ISE 20 % lower, and ITSE 5 % lower, while pole placement attained ITAE 7 % lower; these differences refer to a single experimental realization and are shown below to lie within the run-to-run variability of the prototype. (iii) Simulation versus experiment, for each controller, quantifies the fidelity of the model and is discussed next.

The simulation-to-experiment discrepancies originate from physical effects that are absent from the nonlinear model, and they act differently in the two operating regimes, which is why the two tables must be read separately. Over the full response (Table 3), the simulated indices are the larger ones, and the discrepancy is dominated by the duration of the swing-up: in the absence of friction and backlash, the simulated bang-bang law accumulates more energy per oscillation cycle, so the pendulum overshoots the stabilization cone with higher velocity and is captured later (t ≈ 5.9 s in simulation, against t ≈ 3.5 s in the experiment) and with a larger initial condition, lengthening the transient. Since | e ( t ) | is of the order of π rad during the maneuver, the full-response indices scale with this duration; consistently, the simulation-to-experiment ratio is about 2.3 for ISE and 4.7–4.8 for ITSE, whose time weighting amplifies the large errors occurring later in the slower simulated swing-up, whereas ITAE, dominated by the long stabilization phase, differs by only 12–17 %. Over the stabilization phase (Table 4), the relation is inverted and the experimental indices exceed the simulated ones by factors between 2 and 3, because the real system never rests exactly at the equilibrium: bearing and brush friction, identified on a closely related prototype as F v = 1.2 × 1 0 − 3 N m/(rad/s) and F c = 4.1 × 1 0 − 4 N m by da Silva et al. (2026), together with mechanical backlash, the finite resolution of the incremental encoders (≈ 2.6 × 1 0 − 3 rad for the rod encoder), derivative-filter noise, and the H-bridge dead zone together sustain the small-amplitude oscillation around the upright equilibrium visible in the inset of Figure 5. Measured over the disturbance-free intervals of the stabilization phase, the residual root-mean-square value of σ is 2.7 × 1 0 − 2 to 3.4 × 1 0 − 2 rad in the experiment, against 1.1 × 1 0 − 4 to 1.6 × 1 0 − 4 rad in simulation, a ratio above two orders of magnitude; under the disturbance pulses, by contrast, the same quantity differs by a factor of only 1.4 to 1.5, which indicates that the closed-loop dynamics is well reproduced by the model and that the discrepancy in the integral indices originates almost entirely from this steady-state residual, accumulated over tens of seconds. These observations reinforce the extension of the model to include friction and backlash, indicated as future work in Section 6.

The single realization of Tables 3 and 4 does not allow the modest differences between the controllers to be attributed to the synthesis method rather than to run-to-run variability. To settle this point, a repeatability campaign was carried out on the prototype: fifteen complete swing-up, stabilization, and disturbance runs were executed for each controller, thirty runs in total, under the protocol and disturbance sequence of Equation 31. Table 5 reports the mean and the variance of the stabilization-phase indices over the fifteen runs of each controller.

TABLE 5. Experimental repeatability of the stabilization-phase indices (13–70 s) over fifteen runs per controller.

CV, denotes the coefficient of variation, that is, the ratio between the standard deviation and the mean.

The outcome is conclusive. The difference between the mean indices of the two controllers ranges from 0.6 % (ITAE) to 1.4 % (ITSE), while the coefficient of variation within each controller ranges from 5.6 % to 13.4 %: the difference between synthesis methods is about one order of magnitude smaller than the variation between repetitions of the same method. The 5 % to 20 % differences observed in Table 4 therefore reflect run-to-run variability rather than a systematic advantage of either controller, and the two designs are regarded as equivalent in performance on this prototype. An analogous repeatability analysis of the simulation would not be informative, since the model is deterministic and noise-free and reproduces the same trajectory in every run; the observed dispersion originates precisely from the unmodeled effects discussed before.

The same thirty runs also constitute a substantial experimental robustness test, since each run contains two disturbance events, giving sixty independent rejections applied to the operating closed loop. All sixty events were rejected without loss of stabilization, with a largest angular excursion of 9.90 ° in any event, well inside the ±15° stabilization region. Moreover, the rejections used less than half of the available physical range: the largest control effort observed in any event was 5.78 V of the available 12 V, a margin of 52 %, and the largest reaction-wheel speed was 400 rpm of the 880 rpm motor rating, a margin of 55 %. These margins were obtained on the physical plant, in the presence of the friction, backlash, and quantization effects that the model does not represent.

Finally, the sensitivity of the switching strategy to the stabilization threshold was assessed in simulation on the nonlinear model: the complete closed loop (swing-up, coasting, and stabilization) was simulated with the threshold varied from 4 ° to 45 ° over twenty values, with a finer grid below 10 °, keeping the swing-up boundary at π / 4. Two outcomes are recorded for each case. The time to capture is the instant at which the pendulum angle first enters the stabilization region and remains inside it for at least 5 s, so that a transient crossing is not counted as a capture. Stabilization indicates whether the pendulum was still regulated at the upright equilibrium at the end of the run, that is, whether the capture survived the two disturbance events of Equation 31. The two outcomes are distinct: a run may be captured and later lose stabilization when a disturbance is applied.

Table 6 reports a representative subset. Capture was achieved in every case for both controllers, and stabilization was maintained for every threshold from 6.5 ° to 45 °, with the time to capture varying by only 17 % (5.86–6.83 s) and the stabilization-phase indices by less than 3 % over this entire range, which indicates low sensitivity of the closed loop to the threshold within its operating envelope. Below 6.5 °, the capture still occurs, but the stabilization region becomes narrower than the angular excursion produced by the disturbance (Equation 31), which reaches 6.47 ° in simulation: the pendulum angle then leaves the region during rejection and control is handed back to the swing-up branch, an undesired behavior that causes the pendulum to fall. The adopted threshold of 15 ° leaves the closed loop well away from this boundary. Taking as reference the largest excursion actually measured on the prototype, 9.90 ° over the sixty disturbance events reported above, the adopted threshold provides a margin of 1.5 times.

TABLE 6. Simulation-based sensitivity of the switching strategy to the stabilization threshold (swing-up boundary kept at π / 4; stabilization-phase indices computed over 13–70 s as in Table 4).

The full sweep covers twenty thresholds from 4 ° to 45 °; a representative subset is shown. Capture and stabilization are distinct outcomes: for the 6 ° threshold the pendulum is captured, but stabilization is lost when the first disturbance drives the angle out of the stabilization region, so the indices are not reported.

6 Conclusion

This work compared two state-feedback stabilization strategies, pole placement and the Linear Quadratic Regulator, on a low-cost Reaction-Wheel Inverted Pendulum. A laboratory prototype was built with low-cost components, instrumented with an ESP32 microcontroller, and integrated with MATLAB/Simulink for real-time control and data acquisition. Both controllers were designed using a three-state model obtained from the Lagrangian dynamics of the pendulum coupled with the DC-motor dynamics, linearized about the upright equilibrium, and embedded in a complete pipeline in which a bang-bang energy-injection swing-up raises the pendulum, and an angle-based switching law hands control over to the stabilizing controller. The two strategies were then assessed experimentally under the same operating conditions and disturbance scenario, through the integral indices IAE, ISE, ITAE, and ITSE.

The experiments showed that both controllers stabilize the pendulum at the upright equilibrium on the physical prototype. Over the full response, including the swing-up maneuver, the two controllers perform comparably, with neither one superior across all four indices. Over fifteen experimental runs per controller, the mean stabilization-phase indices differ by 0.6 % to 1.4 % between the two designs, an order of magnitude less than the run-to-run coefficient of variation of 5.6 % to 13.4 %. The two controllers are therefore regarded as equivalent in performance on this prototype, and the choice between them should rest on design considerations, such as the systematic nature of the LQR weighting procedure or the transparency of direct eigenvalue assignment, rather than on measured performance. The sixty disturbance events contained in the same campaign were all rejected without loss of stabilization, using at most 48 % of the available control voltage and 45 % of the rated wheel speed. The simulation results follow the same qualitative behavior as the experiments, and the differences between them are consistent with physical effects not represented in the model, such as friction and mechanical backlash, as quantified in Section 5.3.

The study has limitations. The linearized model does not account for friction, mechanical backlash, or external disturbances, which contributes to the gap between the experimental and simulated responses. The closed-loop eigenvalues of the pole-placement controller and the weighting matrices of the LQR were selected empirically, guided by the actuator and instrumentation constraints discussed in Section 5.1, and the comparison was conducted on a single prototype and at a single operating point. The repeatability campaign was carried out with a single disturbance amplitude and without payload variation, so the reported margins characterize robustness only for the tested scenario.

Future work will extend the mathematical model to include friction and backlash to reduce the gap between simulation and experiment. Further directions include a disturbance-amplitude sweep to determine the largest rejectable amplitude, robustness tests under payload variation and injected sensor noise, and a formal estimate of the region of attraction of the switched swing-up and stabilization loop. Another direction is the use of fuzzy Takagi-Sugeno models to represent the pendulum’s nonlinear dynamics, combined with switched control across the resulting local models, which would allow the upright equilibrium to be addressed without relying on a single linearization (Alves et al., 2016).

Generative AI statement

The author(s) declared that generative AI was used in the creation of this manuscript. Tool used: Claude (Opus 5). Purpose: editorial assistance with the manuscript, comprising suggestions for textual reformulation and improved clarity in English, consistency checking between text, equations, tables, and figures, and standardization of bibliographic references. After using the tool, the authors reviewed and edited the content as needed and assume full responsibility for the final version of the manuscript, including its scientific accuracy. All scientific contributions of this work (mathematical modeling, controller design, simulations, experimental implementation, and interpretation of the results) were conceived and carried out entirely by the authors. The AI tool did not generate experimental data, did not perform autonomous analyses of the results, and is not listed as an author.

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Footnotes

Edited by: Fudong Ge, Tianjin University, China

Reviewed by: Djamel Boucherma, Research Center in Industrial Technologies (CRTIDZ), Algeria

Usman Mohammed, Nile University of Nigeria, Nigeria