Quadruple Tank System (QTS)

• QTS is a lab hydraulic process with four interconnected tanks exhibiting multivariable interactions and nonlinear square-root outflows.
• It serves as a benchmark for testing control strategies like PID, LMPC, and NMPC due to its strong input-output coupling and phase behavior.
• Valve ratios in QTS govern minimum-phase and non-minimum-phase regimes, impacting state estimation and decentralized control performance.

The Quadruple Tank System (QTS) is a laboratory hydraulic process consisting of 4 interconnected water tanks, 2 pumps, and 2 valves, configured as a two-input two-output multivariable plant in which each manipulated input affects more than one tank level through direct flow splitting and cascade drainage. In the formulations reported in recent control and estimation studies, the lower-tank levels are the principal controlled outputs, the upper tanks create internal coupling and inverse-response effects, and the valve ratios determine whether the plant operates in a minimum-phase or non-minimum-phase regime. The QTS is therefore used as a benchmark because it combines strong input-output coupling, nonlinear dynamics, non-minimum-phase behavior for certain valve settings, and outputs that are straightforward to measure but difficult to regulate in a coordinated way (Andersen et al., 2022, Wafi et al., 2023).

1. Physical configuration and benchmark characteristics

The standard QTS layout comprises four tanks arranged in two vertical cascades. Pump 1 sends water to tank 1 and tank 4, while pump 2 sends water to tank 2 and tank 3. Tank 3 drains into tank 1, and tank 4 drains into tank 2. Valve 1 sets the split of pump 1 flow, and valve 2 sets the split of pump 2 flow. This architecture produces a structured but nontrivial multivariable interaction pattern: the lower tanks are directly relevant for regulation, but their dynamics depend on both immediate pump allocation and delayed gravity-driven inflow from the upper tanks (Andersen et al., 2022).

Two related signal conventions appear in the cited literature. In the experimental model-based control studies, the manipulated variables are pump inflows

$$u(t)=\begin{bmatrix}u_1(t) & u_2(t)\end{bmatrix}^\top,$$

the measured outputs are all four tank levels,

$$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$

and the controlled outputs are the bottom-tank levels

$$z(t)=\begin{bmatrix}z_1(t) & z_2(t)\end{bmatrix}^\top.$$

In the decentralized estimation study, the linearized plant is represented with two measured outputs corresponding to the lower tanks,

$$y=\begin{bmatrix}y_1 & y_2\end{bmatrix} =\begin{bmatrix}k_c h_1 & k_c h_2\end{bmatrix},$$

two control inputs given as pump voltages, and four internal states

$$x=\begin{bmatrix}h_1 & h_2 & h_3 & h_4\end{bmatrix}^\top.$$

These are not contradictory descriptions; they reflect different modeling layers and instrumentation choices for the same physical benchmark (Wafi et al., 2023).

The benchmark value of the QTS lies in the simultaneous presence of nonlinear square-root outflows, strong coupling, actuator constraints, and phase behavior that depends on valve settings. This combination makes it suitable for comparing classical SISO loop designs, decentralized multiloop control, linear model predictive control, nonlinear model predictive control, and networked state estimation on both simulated and physical plants (Christensen et al., 14 Sep 2025).

2. Nonlinear dynamics and linearized representations

A recent experimental formulation models the QTS as a continuous-discrete stochastic nonlinear state-space system,

$$dx(t)=f(x(t),u(t),d(t),\theta)\,dt+\sigma(\theta)\,d\omega(t),$$

$y(t_k)=g(x(t_k),\theta)+v(t_k),$

$z(t)=h(x(t),\theta),$

where the state

$$x(t)=\begin{bmatrix}m_1(t)&m_2(t)&m_3(t)&m_4(t)\end{bmatrix}^\top$$

contains tank masses in grams, $d(t)$ represents unknown inflow disturbances, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$0 is a Wiener increment, and $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$1 is measurement noise. For each tank,

$$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$2

with $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$3, outflows

$$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$4

and $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$5. The inflow equations encode both direct pump splitting and the cascade from the upper tanks: $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$6

$$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$7

$$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$8

$$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$9

The controlled-output map is linear in the mass states,

$$z(t)=\begin{bmatrix}z_1(t) & z_2(t)\end{bmatrix}^\top.$$0

with

$$z(t)=\begin{bmatrix}z_1(t) & z_2(t)\end{bmatrix}^\top.$$1

while the measurement map is

$$z(t)=\begin{bmatrix}z_1(t) & z_2(t)\end{bmatrix}^\top.$$2

with

$$z(t)=\begin{bmatrix}z_1(t) & z_2(t)\end{bmatrix}^\top.$$3

This formulation is the basis for both LMPC and NMPC in the experimental comparison studies (Andersen et al., 2022).

A complementary level-based derivation writes the nonlinear mass balances as

$$z(t)=\begin{bmatrix}z_1(t) & z_2(t)\end{bmatrix}^\top.$$4

which yields the familiar four-equation system

$$z(t)=\begin{bmatrix}z_1(t) & z_2(t)\end{bmatrix}^\top.$$5

$$z(t)=\begin{bmatrix}z_1(t) & z_2(t)\end{bmatrix}^\top.$$6

$$z(t)=\begin{bmatrix}z_1(t) & z_2(t)\end{bmatrix}^\top.$$7

$$z(t)=\begin{bmatrix}z_1(t) & z_2(t)\end{bmatrix}^\top.$$8

Linearization around an equilibrium produces either the stochastic deviation model

$$z(t)=\begin{bmatrix}z_1(t) & z_2(t)\end{bmatrix}^\top.$$9

or the deterministic state-space form

$$y=\begin{bmatrix}y_1 & y_2\end{bmatrix} =\begin{bmatrix}k_c h_1 & k_c h_2\end{bmatrix},$$0

In the latter case, the state matrix has leakage terms on the diagonal and upper-to-lower coupling terms of the form $$y=\begin{bmatrix}y_1 & y_2\end{bmatrix} =\begin{bmatrix}k_c h_1 & k_c h_2\end{bmatrix},$$1 and $$y=\begin{bmatrix}y_1 & y_2\end{bmatrix} =\begin{bmatrix}k_c h_1 & k_c h_2\end{bmatrix},$$2, with

$$y=\begin{bmatrix}y_1 & y_2\end{bmatrix} =\begin{bmatrix}k_c h_1 & k_c h_2\end{bmatrix},$$3

The transfer matrix is obtained as

$$y=\begin{bmatrix}y_1 & y_2\end{bmatrix} =\begin{bmatrix}k_c h_1 & k_c h_2\end{bmatrix},$$4

These linearized models support transfer-function analysis, pairing decisions, Kalman filtering, and linear MPC design (Wafi et al., 2023).

3. Valve ratios, transmission zeros, and phase regime

A defining property of the QTS is that its phase behavior depends on how the valves split the pump flows. The decentralized estimation study states the key condition explicitly: if

$$y=\begin{bmatrix}y_1 & y_2\end{bmatrix} =\begin{bmatrix}k_c h_1 & k_c h_2\end{bmatrix},$$5

the system is minimum-phase, whereas if

$$y=\begin{bmatrix}y_1 & y_2\end{bmatrix} =\begin{bmatrix}k_c h_1 & k_c h_2\end{bmatrix},$$6

the system is non-minimum-phase. The physical interpretation given there is equally direct. When $$y=\begin{bmatrix}y_1 & y_2\end{bmatrix} =\begin{bmatrix}k_c h_1 & k_c h_2\end{bmatrix},$$7, more flow goes to the bottom tanks, so the measured outputs are more directly actuated. When $$y=\begin{bmatrix}y_1 & y_2\end{bmatrix} =\begin{bmatrix}k_c h_1 & k_c h_2\end{bmatrix},$$8, more flow is routed through the upper tanks, which makes the lower outputs harder to control and induces inverse-response behavior (Wafi et al., 2023).

The same paper reports two operating points. In the minimum-phase case $$y=\begin{bmatrix}y_1 & y_2\end{bmatrix} =\begin{bmatrix}k_c h_1 & k_c h_2\end{bmatrix},$$9,

$$x=\begin{bmatrix}h_1 & h_2 & h_3 & h_4\end{bmatrix}^\top.$$0

$$x=\begin{bmatrix}h_1 & h_2 & h_3 & h_4\end{bmatrix}^\top.$$1

$$x=\begin{bmatrix}h_1 & h_2 & h_3 & h_4\end{bmatrix}^\top.$$2

so $$x=\begin{bmatrix}h_1 & h_2 & h_3 & h_4\end{bmatrix}^\top.$$3. In the non-minimum-phase case $$x=\begin{bmatrix}h_1 & h_2 & h_3 & h_4\end{bmatrix}^\top.$$4,

$$x=\begin{bmatrix}h_1 & h_2 & h_3 & h_4\end{bmatrix}^\top.$$5

$$x=\begin{bmatrix}h_1 & h_2 & h_3 & h_4\end{bmatrix}^\top.$$6

$$x=\begin{bmatrix}h_1 & h_2 & h_3 & h_4\end{bmatrix}^\top.$$7

so $$x=\begin{bmatrix}h_1 & h_2 & h_3 & h_4\end{bmatrix}^\top.$$8. In the model-based control studies, the nominal setting

$$x=\begin{bmatrix}h_1 & h_2 & h_3 & h_4\end{bmatrix}^\top.$$9

is also reported as giving non-minimum-phase characteristics, which is consistent with the same criterion (Andersen et al., 2022).

Interaction analysis is often expressed through the relative gain array,

$$dx(t)=f(x(t),u(t),d(t),\theta)\,dt+\sigma(\theta)\,d\omega(t),$$0

The decentralized estimation paper states that positive diagonal RGA elements make decentralized control easier, while negative diagonal interaction can destabilize loop pairing. A recurrent misconception is that the QTS is intrinsically non-minimum-phase; the reported results show instead that minimum-phase and non-minimum-phase regimes are both admissible and are governed by valve-ratio selection (Wafi et al., 2023).

4. Identification, disturbance modeling, and state estimation

For offset-free control, the model-based control studies augment the plant with integrating disturbance states,

$$dx(t)=f(x(t),u(t),d(t),\theta)\,dt+\sigma(\theta)\,d\omega(t),$$1

so that the estimator can absorb plant-model mismatch and unknown inflows. Parameters are then identified using a prediction-error method formulated as maximum likelihood. With Gaussian innovations $$dx(t)=f(x(t),u(t),d(t),\theta)\,dt+\sigma(\theta)\,d\omega(t),$$2, the negative log-likelihood is

$$dx(t)=f(x(t),u(t),d(t),\theta)\,dt+\sigma(\theta)\,d\omega(t),$$3

and the estimate is

$$dx(t)=f(x(t),u(t),d(t),\theta)\,dt+\sigma(\theta)\,d\omega(t),$$4

The reported identification results show substantial improvement over nominal parameters. One study gives $$dx(t)=f(x(t),u(t),d(t),\theta)\,dt+\sigma(\theta)\,d\omega(t),$$5 instead of nominal $$dx(t)=f(x(t),u(t),d(t),\theta)\,dt+\sigma(\theta)\,d\omega(t),$$6, $$dx(t)=f(x(t),u(t),d(t),\theta)\,dt+\sigma(\theta)\,d\omega(t),$$7, and modified outlet and tank areas; it reports goodness-of-fit improvement from $$dx(t)=f(x(t),u(t),d(t),\theta)\,dt+\sigma(\theta)\,d\omega(t),$$8 to $$dx(t)=f(x(t),u(t),d(t),\theta)\,dt+\sigma(\theta)\,d\omega(t),$$9 on estimation data and from $y(t_k)=g(x(t_k),\theta)+v(t_k),$0 to $y(t_k)=g(x(t_k),\theta)+v(t_k),$1 on validation data. A later comparison reports the same GOF values and further states that the upper-tank sensor variances were multiplied by 1000 because of poor sensor behavior near turbulence, leading the authors to prefer the lower-tank measurements for parameter estimation (Christensen et al., 14 Sep 2025).

The same later study reports estimated diffusion coefficients

$y(t_k)=g(x(t_k),\theta)+v(t_k),$2

and, for the disturbance-augmented filter, state diffusion coefficients

$y(t_k)=g(x(t_k),\theta)+v(t_k),$3

disturbance diffusion coefficients

$y(t_k)=g(x(t_k),\theta)+v(t_k),$4

and tuned measurement noise variances

$y(t_k)=g(x(t_k),\theta)+v(t_k),$5

These quantities are used in the continuous-discrete Kalman filter and continuous-discrete extended Kalman filter employed by LMPC and NMPC, respectively (Christensen et al., 14 Sep 2025).

A distinct line of work considers distributed state reconstruction. The distributed Luenberger observer is written, for node $y(t_k)=g(x(t_k),\theta)+v(t_k),$6, as

$y(t_k)=g(x(t_k),\theta)+v(t_k),$7

Its local estimation error satisfies

$y(t_k)=g(x(t_k),\theta)+v(t_k),$8

The reported convergence conditions are that the communication graph be connected and undirected and that each local pair $y(t_k)=g(x(t_k),\theta)+v(t_k),$9 be detectable. In the specialized QTS example, two observer nodes are used,

$z(t)=h(x(t),\theta),$0

so each node receives only one bottom-tank output and must rely on neighbor communication for full-state estimation. This directly counters the misconception that one local output is sufficient for complete QTS state reconstruction (Wafi et al., 2023).

5. Control architectures and implementation patterns

The QTS has supported both decentralized and model-predictive control designs. In the experimental comparison, the PID strategy is implemented as two SISO loops with the pairings $z(t)=h(x(t),\theta),$1 for regulation of $z(t)=h(x(t),\theta),$2 and $z(t)=h(x(t),\theta),$3 for regulation of $z(t)=h(x(t),\theta),$4. The transfer matrix is

$z(t)=h(x(t),\theta),$5

and the relevant SISO paths are approximated as

$z(t)=h(x(t),\theta),$6

Skogestad’s IMC rules are then applied with

$z(t)=h(x(t),\theta),$7

$z(t)=h(x(t),\theta),$8

with $z(t)=h(x(t),\theta),$9. The later comparison uses the equivalent SIMC-based structure and gives the saturated PID law with proportional, integral, and derivative terms, derivative filtering with $$x(t)=\begin{bmatrix}m_1(t)&m_2(t)&m_3(t)&m_4(t)\end{bmatrix}^\top$$0, and anti-windup through $$x(t)=\begin{bmatrix}m_1(t)&m_2(t)&m_3(t)&m_4(t)\end{bmatrix}^\top$$1 (Andersen et al., 2022, Christensen et al., 14 Sep 2025).

LMPC and NMPC employ the same state/disturbance-estimation framework but different prediction models. LMPC uses the linearized state-space model, a continuous-discrete Kalman filter, explicit input constraints, and a quadratic objective that penalizes tracking error through

$$x(t)=\begin{bmatrix}m_1(t)&m_2(t)&m_3(t)&m_4(t)\end{bmatrix}^\top$$2

and input-rate change through

$$x(t)=\begin{bmatrix}m_1(t)&m_2(t)&m_3(t)&m_4(t)\end{bmatrix}^\top$$3

The chosen horizon is $$x(t)=\begin{bmatrix}m_1(t)&m_2(t)&m_3(t)&m_4(t)\end{bmatrix}^\top$$4 at sampling time $$x(t)=\begin{bmatrix}m_1(t)&m_2(t)&m_3(t)&m_4(t)\end{bmatrix}^\top$$5, giving $$x(t)=\begin{bmatrix}m_1(t)&m_2(t)&m_3(t)&m_4(t)\end{bmatrix}^\top$$6. In the later notation, $$x(t)=\begin{bmatrix}m_1(t)&m_2(t)&m_3(t)&m_4(t)\end{bmatrix}^\top$$7, $$x(t)=\begin{bmatrix}m_1(t)&m_2(t)&m_3(t)&m_4(t)\end{bmatrix}^\top$$8, and $$x(t)=\begin{bmatrix}m_1(t)&m_2(t)&m_3(t)&m_4(t)\end{bmatrix}^\top$$9. NMPC uses the full identified continuous-discrete nonlinear model, a continuous-discrete extended Kalman filter, the same $d(t)$0, $d(t)$1, and horizon, and a nonlinear optimal control problem transcribed by direct multiple shooting and implemented in CasADi; the later study states that the resulting NLP is solved with IPOPT (Andersen et al., 2022, Christensen et al., 14 Sep 2025).

The decentralized control paper adopts diagonal PI control,

$d(t)$2

with gains obtained by root locus. It reports $d(t)$3 and $d(t)$4 for the minimum-phase regime, and $d(t)$5 and $d(t)$6 for the non-minimum-phase regime. The markedly different tuning constants quantify the increased difficulty of stabilizing the non-minimum-phase case (Wafi et al., 2023).

6. Comparative performance and methodological implications

Performance in the model-based control comparisons is evaluated through the observed tracking error

$d(t)$7

the Normalized Integral Squared Error

$d(t)$8

the Normalized Integral Absolute Error

$d(t)$9

and the Normalized Integral Squared Rate of Movement

$$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$00

In the physical QTS experiment with predefined time-varying setpoints and actuator bounds

$$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$01

the reported results are: PID with NISE $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$02, NIAE $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$03, and NIS$$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$04 $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$05; LMPC with NISE $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$06, NIAE $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$07, and NIS$$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$08 $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$09; and NMPC with NISE $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$10, NIAE $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$11, and NIS$$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$12 $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$13. The paper concludes that LMPC and NMPC outperform PID for the tested time-varying trajectory and have similar overall performance, with NMPC yielding only slightly better tracking at the cost of somewhat larger manipulated-variable rate variation (Andersen et al., 2022).

The later comparison extends this interpretation with simulation studies. When MPC has future setpoint information, the results are PID: NISE $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$14, NIAE $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$15, NIS$$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$16 $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$17; LMPC: $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$18, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$19, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$20; NMPC: $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$21, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$22, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$23. When the same MPCs know only the current setpoint, PID remains at $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$24, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$25, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$26, while LMPC becomes $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$27, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$28, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$29 and NMPC $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$30, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$31, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$32. For deterministic disturbances, the paper reports PID $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$33, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$34, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$35; LMPC $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$36, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$37, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$38; NMPC $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$39, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$40, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$41. For stochastic disturbances, the three controllers are much closer in tracking: PID $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$42, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$43, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$44; LMPC $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$45, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$46, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$47; NMPC $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$48, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$49, $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$50 (Christensen et al., 14 Sep 2025).

Several broader conclusions follow directly from these studies. First, a common assumption that NMPC should decisively dominate LMPC is not supported in the tested QTS regimes; the reported explanation is that the linearized identified model is already sufficiently accurate, and the QTS nonlinearities do not strongly separate LMPC and NMPC under the chosen operating conditions (Andersen et al., 2022). Second, a common assumption that MPC is intrinsically superior to PID also requires qualification: in the later study, the primary advantage of MPC is its ability to use information about future setpoints, and without such preview decentralized PID can outperform MPC in tracking (Christensen et al., 14 Sep 2025). Third, the decentralized estimation study shows that minimum-phase and non-minimum-phase QTS regimes are not merely theoretical variants but materially different control problems, with the non-minimum-phase case requiring roughly ten times longer settling time under the reported decentralized PI design (Wafi et al., 2023).

Recent implementations underscore that these conclusions were obtained on more than a purely abstract benchmark. The physical experiments were implemented in Python, interfaced through OPC UA, executed at $$y(t_k)=\begin{bmatrix}y_1(t_k) & y_2(t_k) & y_3(t_k) & y_4(t_k)\end{bmatrix}^\top,$$51 s, and recorded in an SQL or PostgreSQL database rather than through commercial control software. This suggests that the QTS remains valuable not only as a compact nonlinear MIMO model, but also as a reproducible platform for testing estimation, preview-based control, multivariable constraint handling, and networked observer design under laboratory conditions (Andersen et al., 2022, Christensen et al., 14 Sep 2025).