A cascade active disturbance rejection control based on SPSA-memory optimization for SCR denitration process

Modeling of SCR denitration process

SCR is a post-combustion denitration method as shown in Figure 1, which that first achieved commercial operation in the late 1960s. It utilizes reducing agents to selectively react with NO x in the presence of metal catalysts to produce N 2 and H 2 O . The reducing agent (ammonia solution) is transported from the storage system to the vaporizer, where it is atomized, mixed with high-temperature flue gas, and diluted to a safe concentration (less than 5% by volume) before being injected into the inlet flue of the SCR reactor. The relevant chemical reduction reactions are as follows

4 NO + 4 NH 3 + O 2 → 4 N 2 + 6 H 2 O 2 NO 2 + 4 NH 3 + O 2 → 3 N 2 + 6 H 2 O

(1)

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Figure 1.

SCR denitration process.

There are many performance indicators affecting the denitration efficiency of SCR, primarily: flue gas flow rate, inlet/outlet NO x concentration, catalyst bed pressure drop, and temperature. In the design of an SCR system, the main performance indicators considered are the NO x removal rate, ammonia slip, and device availability. The main task of coal-fired SCR denitration control is to ensure that the emission concentrations of nitrogen oxides and ammonia slip remain within the limits permitted by national standards.

The outlet NO x concentration is dynamically regulated by the ammonia injection rate. As the core control parameter, this injection rate is adjusted via the ammonia injection valve opening. Literature (Hou et al., 2020) indicates that the process dynamics within an SCR reactor can generally be characterized by a first- or second-order model with time delay. Given the measurement delay associated with ammonia flow and outlet NO x concentration, this relationship is best described by a second-order inertial element combined with a pure time delay. Conversely, since changes in valve opening directly affect the ammonia flow rate, this dynamic can be represented by a first-order inertial element without delay. Consequently, the simplified SCR denitration model is approximated as a third-order system with significant time delay

G ( s ) = Y ( s ) U ( s ) = K ( T 1 s + 1 ) ( T 2 s + 1 ) 2 e − τs

(2)

Here, U ( s ) represents the ammonia injection rate (kg/h), and Y ( s ) denotes the SCR system’s outlet NO x concentration ( mg/Nm 3 ). Using real-world operating data from a 300 MW load condition at China Southern Power Grid, we employed the PSO algorithm to identify the model parameters. Specifically, the time constants T 1 and T 2 , the delay time constant τ, and the gain K within the transfer function were determined, yielding

G ( s ) = 2 . 926 ( 1 . 4 s + 1 ) ( 179 s + 1 ) 2 e − 124 s

(3)

Analyses of control difficulties

The SCR denitration model is characterized by significant time delays, making traditional control methods such as PID, sliding mode control (SMC), and standard ADRC inadequate. In practical power generation, maintaining safety necessitates strictly zero overshoot during the control process—a requirement that traditional methods often fail to meet without significantly compromising response speed. Furthermore, frequent load variations cause substantial shifts in plant dynamics. Consequently, utilizing fixed parameters leads to performance degradation, and offline parameter optimization techniques are ill-suited for handling real-time process variations.

This study proposes the inclusion of a delay compensation module in the controller to effectively mitigate the performance degradation associated with large time delays. Additionally, an SPSA module is incorporated to facilitate online parameter optimization. This allows for aggressive parameters during large deviations to facilitate swift tracking, while adopting conservative parameters during small deviations to ensure zero overshoot. Consequently, the system adaptively accommodates load fluctuations and real-time operational conditions. This autonomous approach obviates the need for manual intervention and prevents frequent startup and shutdown cycles of thermal power units, thereby enhancing overall system efficiency.

Principle of ADRC

ADRC, originally proposed by J. Han, is a robust control paradigm that operates independently of precise model parameters. Its fundamental philosophy involves treating internal model uncertainties and external environmental disturbances as a lumped “total disturbance.” By estimating and compensating for this total disturbance in real time, ADRC effectively transforms an uncertain nonlinear plant into a canonical chain of integrators. Building upon this, LADRC employs a bandwidth-parameterization technique, simplifying complex tuning into the adjustment of a single bandwidth parameter. A typical second-order LADRC system comprises three core components: a second-order linear tracking differentiator (LTD), a third-order linear extended state observer (LESO), and a linear state error feedback (LSEF) control law.

The LESO serves as the cornerstone of the LADRC framework. It formulates the “total disturbance”—which encompasses both internal model uncertainties and external interferences—as an extended state variable. By processing system input–output data through linear gains, the LESO achieves real-time estimation of both the nominal system states and this extended state.

The parameters in the equation are explained as follows:y is the actual output value,u is the control quantity given by the LSEF control rate,w is the disturbance of the plant, z 1 is the observed value of the actual output y, z 2 is the observed value of the differential of the actual output, z 3 is the estimated value of the disturbance, and b 0 is an estimate of the input gain.

Suppose that the model of the controlled object is

ÿ = bu + f ( y , u , w )

(4)

where y is the output of the system,u is the control quantity,b is the high-frequency gain of the system, and f is the generalized disturbance. Taking x 1 = y , x 2 = ẏ , the generalized disturbance f is defined as the extended state of the system x 3 , and h = ẋ 3 . The state-space representation is

{ ẋ 1 = x 2 ẋ 2 = b 0 u + x 3 ẋ 3 = h y = x 1

(5)

The third-order linear ESO is designed as follows

{ e = z 1 − y ż 1 = z 2 − β 1 e ż 2 = b 0 u + z 3 − β 2 e ż 3 = − β 3 e

(6)

where e is the observation error; z 1 , z 2 , and z 3 are the estimates of the output y; its derivative ẏ; and the total disturbance f, respectively. b 0 is the estimate of the system input gain. The observer gains are selected as β 1 = 3 ω o , β 2 = 3 ω o 2 , and β 3 = ω o 3 based on bandwidth parameterization (Gao, 2003).

The disturbance compensation control law is

u = u 0 − z 3 b 0

(7)

where u 0 is the virtual control signal generated by the LSEF law.

IMC cascade active disturbance rejection controller

Controlling high-order systems with significant delays presents a complex engineering challenge. For such systems, the disparity in observer gains becomes significant, making it difficult to balance the convergence speed of each state estimate in a standard high-order ESO. Furthermore, plant parameters vary with operating conditions, leading to performance degradation. To address this, we decompose the plant dynamics into a nominal part G 1 ( s ) and an uncertain part G 2 ( s )

G ( s ) = G 1 ( s ) × G 2 ( s )

(8)

where G 1 ( s ) represents the fixed, deterministic model of the plant, and G 2 ( s ) represents the time-varying dynamics and uncertainties.

An IMC is designed for the nominal model G 1 ( s ) and placed in the inner loop to enhance the precision and speed of the overall control system. The continuous variation of G 2 ( s ) due to load changes is treated as an external disturbance, which is estimated and compensated for by the LESO in the outer loop. This cascade structure significantly enhances adaptability. The structure of the IMC-based cascade ADRC is shown in Figure 2.

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Figure 2.

IMC cascade ADRC structure.

In the IMC block, Ĝ 1 ( s ) is the internal model approximating G 1 ( s ) . The internal model controller G IMC ( s ) is designed as

G IMC ( s ) = G filter ( s ) Ĝ 1 ( s )

(9)

where Ĝ filter ( s ) is a low-pass filter added to ensure causality and robustness. In this architecture, the outer loop controller compensates for model variations and external disturbances, while the inner loop controller ensures precise tracking of the nominal model. This synergy significantly enhances both tracking performance and disturbance rejection capabilities.

IMC-based cascade ADRC with delay compensation

Delay compensation techniques are employed to mitigate the adverse effects of significant delays, transforming the effective open-loop response into a more manageable form. This approach reduces the computational burden on the LESO while concurrently enhancing observation precision. The fundamental concept involves reducing the observer’s sensitivity to measurement delays through a specialized compensation structure, as illustrated in Figure 3.

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Figure 3.

Block diagram of the delay compensation module.

The design principle is as follows: when the input reference changes rapidly (i.e., high-frequency components dominate), the feedback signal y rt should rely more on the model output y m (which has no delay) to ensure system stability. Consequently, the weighting term F 1 ( s ) 1 + F 2 ( s ) should have a larger magnitude. Conversely, as the system approaches a steady state, the contribution of the actual system output y p (which contains the delay) must be increased to minimize steady-state error. In this scenario, F 1 ( s ) 1 + F 2 ( s ) should approach zero. The weighting coefficients are designed as follows

{ F 1 ( s ) = λ 1 τ max ⁡ s = λ 1 ( τ 0 + τ 1 ) s , λ 1 > 0 F 2 ( s ) = λ 2 τ max ⁡ s = λ 2 ( τ 0 + τ 1 ) s , λ 2 > 0 F 3 ( s ) = λ 3 τ min ⁡ s = λ 3 τ 0 s , λ 3 ≥ 0

(10)

This leads to the transfer relationships

1 + F 3 ( s ) 1 + F 2 ( s ) = 1 + λ 3 τ 0 s 1 + λ 2 ( τ 0 + τ 1 ) s

(11)

F 1 ( s ) 1 + F 2 ( s ) = λ 1 ( τ 0 + τ 1 ) s 1 + λ 2 ( τ 0 + τ 1 ) s

(12)

Consider the plant described by equation (2). The time delay satisfies τ ∈ [ τ min ⁡ , τ max ⁡ ] , where τ min ⁡ = τ 0 and τ max ⁡ = τ 0 + τ 1 . The controller parameters are selected as λ 1 = 2 , λ 2 = 0 . 1 , and λ 3 = 1 , with G 0 ( s ) serving as the boundary model of the plant. By incorporating a delay compensation mechanism into the IMC-based cascade ADRC framework, the complete control structure is obtained, as illustrated in Figure 4.

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Figure 4.

Structure of IMC-based cascade ADRC with delay compensation.

Principle of SPSA algorithm

The stochastic approximation (SA) algorithm (Chin, 1997; Robbins and Monro, 1951), initially introduced by Robbins and Monro (1951), is an optimization method based on stochastic gradient descent. It is primarily utilized to address optimization challenges in online and incremental learning scenarios. Its fundamental premise revolves around iteratively refining parameter estimates via gradient descent. Unlike conventional gradient descent techniques, SA selectively incorporates information from a subset of samples during each iteration, thereby avoiding the computational burden of processing the entire dataset. The basic update rule for the SA algorithm is

θ n + 1 = θ n − α n g n ( θ n )

(13)

Here, θ n represents the parameter vector at the n th iteration, α n denotes the learning rate, and g n ( θ n ) signifies the gradient estimate.

The SPSA algorithm is a refined iteration of SA, tailored to effectively address high-dimensional and nonlinear optimization challenges. Its foundational premise entails introducing random perturbations to all parameters simultaneously and estimating the gradient based on function values post-perturbation. The SPSA iterative update law is defined as

θ k + 1 = θ k − a k ĝ k ( θ k )

(14)

where θ k represents the parameter vector at the k th iteration, ĝ k ( θ k ) represents the estimated gradient, and a k is the learning rate. A random perturbation vector Δ k = [ Δ k 1 , … , Δ kp ] T is introduced. The components Δ ki are independent and identically distributed (i.i.d.) symmetric zero-mean random variables, satisfying | Δ ki | ≤ α 0 and | 1 ∕ Δ ki | ≤ α 1 for all k, where α 0 , α 1 ∈ ℝ + .

At the heart of the SPSA algorithm lies the simultaneous perturbation mechanism. For each iteration, two measurements of the loss function J ( ⋅ ) are taken at perturbed points θ k + c k Δ k and θ k − c k Δ k , where c k is a scaling parameter. The gradient is then estimated as follows

ĝ k ( θ k ) = [ f ( θ k + c k Δ k ) − f ( θ k − c k Δ k ) 2 c k Δ k 1 ⋮ f ( θ k + c k Δ k ) − f ( θ k − c k Δ k ) 2 c k Δ kp ] = f ( θ k + c k Δ k ) − f ( θ k − c k Δ k ) 2 c k [ Δ k 1 − 1 ⋮ Δ kp − 1 ]

(15)

The calculated result is substituted into equation (14) to update the parameters. This perturbation-driven methodology effectively circumvents challenges inherent in high-dimensional problems by requiring only two function evaluations per iteration, regardless of the dimension of the parameter vector.

SPSA-based parameter optimization for IMC-ADRC

In the proposed IMC-based cascade ADRC system, the inner loop IMC, the outer loop LSEF controller, and the LESO involve multiple parameters that require optimization. Based on the bandwidth parameterization technique for linear ADRC, the system performance is primarily determined by three critical parameters: the controller bandwidth ( ω c ), the observer bandwidth ( ω o ), and the IMC filter time constant ( T f ). These parameters play distinct roles in the control system. Specifically, the observer bandwidth ω o determines the convergence speed of the ESO; an excessively large ω o may amplify noise and cause oscillations, whereas a small ω o results in poor disturbance tracking. The controller bandwidth ω c governs the response speed of the closed-loop system; a higher ω c yields a faster response but may induce overshoot or instability. For the inner loop, the filter time constant T f serves as the primary tuning parameter affecting model matching and robustness.

Dynamic adjustment of these parameters in response to tracking errors can significantly enhance control performance. During periods of large error, parameters can be tuned for aggressive tracking, while near the setpoint, they can be adjusted to prevent overshoot. Furthermore, the SPSA algorithm enables the system to adapt rapidly to the load variations inherent in SCR denitration processes.

When tuning the controller parameters using the SPSA algorithm, it is necessary to estimate the order of magnitude of the parameters and establish the upper and lower limits for parameter adjustments. The learning rate a k and scale parameter c k are determined based on the order of magnitude of the parameters, as shown in equations (16) and (17).

a k = a ( k + A ) α

(16)

c k = c k γ

(17)

where a , c , α , and γ are positive constants. Following standard SPSA guidelines (Spall, 1998), the theoretical values are typically set to α = 0 . 602 and γ = 0 . 101 . In environments with significant noise, the value of a may be reduced and c increased to enhance stability. The stability constant A is typically set to approximately 10% of the maximum number of iterations allowed.

Let θ = [ ω c , ω o , T f ] T denote the parameter vector. The perturbation vectors are generated as θ k + = θ k + c k Δ k and θ k − = θ k − c k Δ k , where Δ k is the random perturbation vector. The performance index (cost function) is defined as the squared tracking error

J ( θ ) = 1 2 ( r − y ) 2

(18)

where y is the system output and r is the reference setpoint. The schematic of the online parameter optimization process is illustrated in Figure 5.

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Figure 5.

Flowchart of the SPSA-based online parameter optimization.

Assuming noise-free measurements for the purpose of theoretical derivation, the parameter update law is given by θ k + 1 = θ k − a k ĝ k ( θ k ) , where the gradient estimate ĝ k ( θ k ) is computed as follows

ĝ k ( θ k ) = J ( θ k + ) − J ( θ k − ) 2 c k [ Δ k 1 − 1 ⋮ Δ kp − 1 ]

(19)

The bias of this gradient estimate is analyzed as follows. Let F k denote the sigma-algebra generated by the history of iterations up to k. The conditional expectation of the l th component of the gradient estimate is as follows

E [ ĝ kl ( θ k ) | F k ] = E [ J ( θ k + c k Δ k ) − J ( θ k − c k Δ k ) 2 c k Δ kl | F k ]

(20)

Expanding the cost function J ( θ ) using a third-order Taylor series around θ k yields

J ( θ k ± c k Δ k ) = J ( θ k ) ± c k g ( θ k ) T Δ k + 1 2 c k 2 Δ k T H ( θ k ) Δ k ± 1 6 c k 3 J ‴ ( θ ¯ k ± ) Δ k ⊗ 3

(21)

where H ( θ ) is the Hessian matrix, Δ k ⊗ 3 denotes the tensor product, and θ ¯ k ± lies on the segment between θ k and θ k ± c k Δ k . Assumption 1: The cost function J ( θ ) is three times continuously differentiable, and its third-order derivative is bounded, i.e., | J ‴ ( θ ) | ≤ α 2 . Substituting equation (21) into equation (20), the quadratic terms cancel out, leading to

E [ ĝ kl ( θ k ) | F k ] = E [ g ( θ k ) T Δ k Δ kl | F k ] + c k 2 12 E [ ( J ‴ ( θ ¯ k + ) + J ‴ ( θ ¯ k − ) ) Δ k ⊗ 3 Δ kl | F k ]

(22)

The first term simplifies to g l ( θ k ) because the perturbation components Δ ki are i.i.d. symmetric zero-mean variables, implying E [ Δ ki ∕ Δ kl ] = 0 for i ≠ l . Consequently

E [ ĝ k ( θ k ) | F k ] = g ( θ k ) + O ( c k 2 )

(23)

Since c k → 0 as k → ∞ , the bias term vanishes. This proves that the SPSA gradient estimate is asymptotically unbiased, ensuring the validity of the optimization process. When the SPSA algorithm is applied to the real-time control process of IMC cascade active disturbance rejection controller, the complete online optimization algorithm flow is shown in Figure 6.

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Figure 6.

Online parameter optimization of SPSA algorithm.

SPSA-memory optimization

When the load of the SCR denitration process fluctuates, relying solely on the most recent parameter estimates for continuous optimization may lead to slow convergence, particularly when the new operating point deviates significantly from the previous state. This lag in adaptation can severely degrade transient control performance.

To address this, this study proposes a parameter resetting mechanism triggered by load changes. Since load transitions are frequent in practical operation, the optimal parameters obtained from previous SPSA iterations can be stored and utilized as optimal initial guesses for subsequent transitions. To facilitate this, a memory bank (look-up table) is constructed to store optimal control parameters corresponding to different operating conditions. Upon the detection of a load transition, the system retrieves the recommended parameters corresponding to the new operating range from the memory bank to initialize the controller. Subsequently, building upon these initial values, the SPSA algorithm further fine-tunes the parameters via stochastic perturbations and gradient descent, ensuring convergence to the precise optimum for the current operating conditions. This approach ensures that the controller initiates from an appropriate starting point, thereby mitigating the learning lag typically associated with standard SPSA, significantly shortening the initial parameter searching distance, and accelerating the tracking speed. Furthermore, the memory bank is dynamically updated whenever the SPSA algorithm converges to a parameter set that yields superior performance metrics compared to the currently stored values.

Tracking performance simulation

To accurately simulate the complex time-varying load conditions of the SCR denitration process, a timer was integrated into the simulation process to regulate load variations. The system was initialized at a load of 300 MW. Subsequently, the load was increased to 400 MW at 8000 s and further to 500 MW at 24,000 s. Consequently, the system’s transfer function evolved in response to these load shifts. This simulation compares the performance of four controllers: PID, ADRC, IMCADRC, and the memory-based SPSA-optimized IMCADRC (mspsaIMCADRC). Figure 7 displays the step response curves for each controller.

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Figure 7.

Step response curves for four controllers.

To quantitatively assess the effectiveness of the control strategies, Table 1 presents detailed performance metrics for the four controllers during step response and disturbance rejection tests. Regarding setpoint tracking performance, the mspsaIMCADRC exhibits remarkable superiority. It achieves a rise time ( t r ) of only 573.7 s, and the settling time ( t s , defined by a ± 2 % error band) is reduced to 1119.2 s. This represents an approximate threefold increase in response speed compared to traditional PID and ADRC controllers, while maintaining negligible steady-state error.

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Table 1.

Quantitative performance comparison of four controllers in setpoint tracking and disturbance rejection.

Controller	Rise time ( t r /s)	Settling time ( t s /s)	Steady-state error	Max. dynamic deviation	Recovery time ( t rec /s)
mspsaIMCADRC	573.7	1119.2	0.00000	0.2254	2633.9
IMCADRC	1006.0	2163.9	0.00000	0.2404	3277.6
ADRC	1846.8	3337.5	0.00001	0.2873	5348.1
PID	1725.4	3599.7	0.00022	0.2299	4376.4

In terms of disturbance rejection, the analysis focused on the system’s performance following the sudden load change at 8000 s. Subject to load fluctuations, all controllers demonstrated varying degrees of dynamic deviation. Notably, the mspsaIMCADRC suppressed the maximum dynamic deviation to 0.2254, marginally outperforming the other controllers. More significantly, the proposed method demonstrated a substantial advantage in recovery speed. With a recovery time of just 2633.9 s, it effectively halved the duration required by traditional algorithms. These findings strongly confirm that integrating MSPSA online optimization with an internal model structure significantly enhances both system rapidity and robustness under variable operating conditions.

Figure 8 further elucidates the specific contribution of the memory module within the optimization algorithm. While standard SPSA optimization enhances performance during the initial ascent, it often induces a response delay during load switching (Ai et al., 2024), as the algorithm persists in adjusting parameters based on prior values. This delay extends the time required for the system to track reference values. By incorporating a memory function, the MSPSA algorithm retrieves and resets parameters corresponding to the current load condition using historical data. Simulation results indicate that with the inclusion of the memory function, the maximum dynamic deviation induced by disturbances is reduced by 6.19%, and the time to recover to a steady state is shortened by 13.04%. Consequently, the update process during load switching becomes smoother, leading to further improvements in tracking performance. The memory-based approach eliminates the delay associated with re-learning, ensuring rapid adaptation to new operating conditions without overshoot.

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Figure 8.

Impact of memory module on tracking and disturbance rejection.

Furthermore, Figure 9 records the variation of the control input during the simulation to evaluate the actuator-friendliness of the proposed strategy in practical industrial applications. As illustrated, the control effort generated by the MSPSA-IMCADRC remains remarkably smooth and safely bounded throughout the entire operation. Even under severe load transitions at 8000 and 24,000 s, the controller executes rapid yet continuous adjustments without inducing hazardous overshoots or oscillatory behavior in the control signal. This smooth and stable control trajectory conclusively demonstrates that the proposed algorithm satisfies the stringent actuator constraints of real-world thermal power units, thereby ensuring prolonged equipment lifespan and safe operational continuity.

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Figure 9.

Trajectories of the control input under varying load conditions.

Stability analysis

In the control structure shown in Figure 4, the inclusion of the IMC in the inner loop effectively compensates for the nominal dynamics G 1 ( s ) . Consequently, the LESO and LSEF in the outer loop can be designed based on the simplified generalized plant. The frequency characteristics of the IMC controller, given by equation (30), are analyzed below. The magnitude response A ( ω ) and phase response ϕ ( ω ) are derived as follows

A ( ω ) = | G IMC ( jω ) | = ( 1 . 4 ω ) 2 + 1 1 . 1 2 ( ( T f ω ) 2 + 1 )

(31)

ϕ ( ω ) = ∠ G IMC ( jω ) = arctan ⁡ ( 1 . 4 ω ) − arctan ⁡ ( 1 . 1 T f ω )

(32)

Since G IMC ( s ) is integrated in series within the closed-loop configuration, it is evident from equations (31) and (32) that the inclusion of G IMC ( s ) tends to increase the gain margin (GM) while potentially reducing the phase margin (PM). The filter time constant T f serves as a tuning knob: a smaller T f can be selected to preserve the PM. Building upon this foundation, the delay compensation module is incorporated. The open-loop Bode diagram of the system under a 300 MW load condition is analyzed to verify stability.

To assess robustness, the time constant of the inertial element in the plant was varied by ± 20 % . The Bode diagrams are presented in Figure 10. Under the nominal 300 MW load (equation (24)), the system exhibits a gain crossover frequency ω gc of 0.0152 rad/s with a PM of 48 . 2 ∘ , and a GM of 200 dB (indicating high stability at low frequencies). When the model parameters (time constants) increase by 20 % (equation (33)), the system exhibits ω gc = 0 . 0144 rad ∕ s , a GM of 204 dB, and a PM of 48 . 4 ∘ . Conversely, when the model parameters decrease by 20 % (equation (34)), the system exhibits ω gc = 0 . 0165 rad ∕ s , a GM of 194 dB, and a PM of 48 . 4 ∘ .

G ( s ) + 20 % = 2 . 926 ( 1 . 4 s + 1 ) ( 215 s + 1 ) 2 e − 149 s

(33)

G ( s ) − 20 % = 2 . 926 ( 1 . 4 s + 1 ) ( 143 s + 1 ) 2 e − 99 s

(34)

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Figure 10.

Bode diagram of the system when the load is 300 MW: (a) the amplitude margin and phase angle margin of the system under 300 MW load; (b) amplitude margin and phase margin after the time constant is increased by 20%; (c) amplitude margin and phase margin after the time constant is reduced by 20%.

The minimal variation in PM ( 48 . 2 ∘ to 48 . 4 ∘ ) under significant parameter uncertainty confirms the strong robustness of the proposed control scheme.

SPSA parameter optimization procedure

In the simulation, the SPSA algorithm achieves online parameter optimization for the IMC-based cascade ADRC. The trajectory of the three key parameters ( ω c , ω o , T f ) during the adaptation process is illustrated in Figure 11. It can be observed that during the transient rise phase and the stabilization period following load transitions, the parameter trajectories exhibit significant variation, indicating active adaptation to minimize tracking error. Once the system reaches steady state, the parameters converge to stable values, demonstrating the algorithm’s capability to quickly locate optimal settings.

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Figure 11.

Trajectories of controller parameters during SPSA online optimization: (a) active disturbance rejection controller bandwidth ω c update process; (b) active disturbance rejection observer bandwidth ω o update process; (c) IMC filter time constant update process.