Revolutionizing Hydraulic Control: A Cutting-Edge State-Feedback Sliding-Mode Controller for Turbine Governance

Controller design methodology

The HTGS model, which takes into account the uncertainty of the control signal of the dynamic system alongside external input disturbances, is detailed by Eq. (17). This study integrates equivalence-based SMC principles with auxiliary feedback-based SMC analysis to construct a SFRSMC in HTGS.

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In this research, we present the concept of the sliding surface (varvec{mathscr {S}}) for the HTGS. The sliding surface is connected to the system state vector (varvec{mathscr {X}}) and is defined as follows:

where (varvec{B}_u^T) denotes the transpose of the control input matrix, and (varvec{mathscr {P}}) is a positive definite matrix. The selection of the positive definite matrix (varvec{mathscr {P}}) is crucial for achieving the desired control attributes within the HTGS.

By judiciously designing the sliding matrix (varvec{mathscr {P}}), we can ensure that the sliding surface (varvec{mathscr {S}}) exhibits desirable behavior, leading to effective regulation and stabilization of the HGU. The ultimate objective is to attain (varvec{mathscr {S}}rightarrow 0) as the system state vector (varvec{mathscr {X}}) converges to the target state, thereby ensuring accurate regulation and stability in the HTGS.

For the HTGS, the mathematical representation of the SFRSMC formulated in this study can be articulated as:

where (u_{eq}) represents the equivalence-based control regulation; (u_n) denotes the robust control regulation.

According to the equivalence control principle, and by assuming (Delta u = 0), we can derive from Eq. (16) and (dot{varvec{mathscr {S}}}=0) the following relationships:

Consequently, the equivalence-based control regulation (u_{eq}) can be determined as follows:

Furthermore, based on the design principle of SMC, to ensure that (varvec{S}dot{varvec{S}} le 0), the robust control regulation (u_n) included in the controller output (varvec{u}) is delineated as follows:

where (varepsilon _0) signifies the switching gain; (operatorname {sgn}(*)) refers to the sign function.

Subsequently, the dynamics of the sliding surface (varvec{mathscr {S}}) is examined through Lyapunov’s stability theorem, and the Lyapunov function is selected as:

The derivative of the sliding surface (varvec{mathscr {S}}) is obtained:

Then, the derivative of the Lyapunov function (varvec{V}_{S}) defined by Eq. (23) is derived:

According to Lyapunov’s stability theorem, when (dot{varvec{V}}_S le -varepsilon _0 |varvec{mathscr {S}}|), as (trightarrow infty), it becomes possible to ensure (varvec{mathscr {S}}rightarrow 0).

In conclusion, the mathematical representation of the SFRSMC for the HTGS is formulated by integrating Eq. (21) and Eq. (22):

In this formula, a combination of terms dictates the control input u. The initial component originates from the equilibrium-based control law (u_{eq}). This element guarantees that the control strategy directs the system toward the intended sliding surface and monitors the target state. The subsequent segment (u_{n}) involves the sliding-mode robust control characteristic of the SFRSMC approach. It implements a sliding mode technique to enhance resilience against uncertainties and disturbances within the HTGS. The final control law represented by Eq. (26) offers a practical and robust method for managing the HGU and attaining stability and precise control during HTGS operations.

SMC analysis of auxiliary feedback-based

The preceding analysis illustrates that the construction of the symmetric positive definite matrix (varvec{mathscr {P}}) significantly influences the performance of state feedback control. In this manuscript, we refer to? for the design concept of SMC analysis based on auxiliary feedback. Building upon this idea, we creatively introduce the combined (H_2/H_{infty }) LMI to formulate the sliding matrix (varvec{mathscr {P}}) so that the controller achieves optimal regulation and superior robustness performance. Initially, the mathematical representation of the controller is reformulated as:

where (varvec{upsilon }=-varvec{Okay}varvec{mathscr {X}}+u_{eq}+u_{n}).

By integrating the rewritten controller provided by Eq. (27) into the state-space equations laid out by Eq. (17), the closed-loop system can be expressed as follows:

where (varvec{Sigma }_d=[begin{array}{cc} varvec{B}_{u}&varvec{B}_{d} end{array}]); (varvec{d}^{*}=left[ begin{array}{c} varvec{upsilon }+Delta u d end{array}right]).

For the dynamic system characterized by Eq. (28), two sets of control performance output vectors are included, one for the (H_2) control performance output vector and another for the (H_{infty }) control performance output vector, leading to the following augmented dynamic system:

where (varvec{Z}_{2} in mathbb {R}^{9 occasions 1}), (varvec{Z}_{infty } in mathbb {R}^{9 occasions 1}) represent the (H_2) and (H_{infty }) control performance output vectors, respectively.

Within this augmented system, we have two control performance output vectors (varvec{Z}_{2}) and (varvec{Z}_{infty }), which are affected by the matrices (varvec{C}_{infty }), (varvec{D}_{infty 1}), (varvec{D}_{infty 2}), (varvec{C}_2) and (varvec{D}_{2}), respectively. The augmented dynamic system, defined by Eq. (29), allows us to examine and design the controller based on varying control performance criteria, specifically, (H_2) and (H_{infty }) control performance, providing a comprehensive framework for managing and enhancing the HTGS regarding these performance objectives. In this research, the weighting matrices (varvec{C}_{infty }), (varvec{D}_{infty 1}), (varvec{D}_{infty 2}), (varvec{C}_2), and (varvec{D}_{2}) in Eq. (29) are specified as:

These matrices (varvec{C}_{infty } in mathbb {R}^{9 occasions 8}), (varvec{D}_{infty }in mathbb {R}^{9 occasions 1}), (varvec{C}_2 in mathbb {R}^{9 occasions 8}), and (varvec{D}_{2} in mathbb {R}^{9 occasions 1}) represent the structured coefficient matrices of the appropriate dimensions; it should be noted that (varvec{Q}_{infty }), (varvec{R}_{infty }), (varvec{R}_ {infty }), (varvec{Q}_{2}), and (varvec{R}_{2}) are defined in this manuscript as:

where ({ q}_i ge)0 ((i=1), (cdots), 8) and ({ r}) denote the weighting factors.

According to the sliding surface (varvec{mathscr {S}} = varvec{B}_u^Tvarvec{mathscr {P}}varvec{X}) outlined in this document, the matrix (varvec{mathscr {P}}) primarily functions to organize and integrate the state feedback information of the regulated system. Based on this premise, this paper delineates the association between the state feedback matrix (varvec{Okay})
and the constructive specific matrix (varvec{mathscr {P}}) in (28) is articulated as (varvec{Okay} = varvec{W}varvec{mathscr {P}}). The matrix (varvec{mathscr {P}}) is determined by crafting a suitable matrix (varvec{Okay}) such that the governed entity described by Eq. (28) exhibits combined (H_2/H_{infty }) management efficiency. The combined (H_2/H_{infty }) management efficiency necessitates that the closed-loop system fulfills the following efficiency criteria:

1) (H_{infty }) robustness efficiency, the closed-loop switch functionality (widetilde{varvec{G}}_{z_{infty }d^{*}}(s)) from (d^{*}) to (varvec{Z}_{infty }) must satisfy:

The (H_{infty }) efficiency via a suggested achieving matrix (varvec{Okay}_{infty } = varvec{W}_{infty }varvec{mathscr {P}}_{infty }) is realized if and only if the matrices (varvec{W}_{infty }) and (varvec{mathscr {P}}_{infty }) exist, and the subsequent LMI is satisfied:

where (langle * rangle _s=(*)+(*)^T).

2) (H_2) dynamic response efficiency, the closed-loop switch functionality (widetilde{varvec{G}}_{z_{2}d^{*}}(s)) from (d^{*}) to (varvec{Z}_{2}) must fulfill:

The (H_{infty }) efficiency via a suggested achieving matrix (varvec{Okay}_2 = varvec{W}_2varvec{mathscr {P}}_2) is realized if and only if the matrices (varvec{W}_{2}), (varvec{mathscr {P}}_{2}), and (varvec{mathscr {Q}}) exist and the subsequent LMI is satisfied:

where (textbf{tr}(varvec{N})) denotes the trace of matrix (varvec{N}).

For simplicity, let us denote (mathscr {P}_{infty }=varvec{mathscr {P}}_2 triangleq varvec{mathscr {P}}) and (varvec{W}_{infty }=varvec{W}_2 triangleq varvec{W}). The closed-loop suggested achieving matrix (varvec{Okay}=varvec{W}varvec{mathscr {P}}) must simultaneously meet the efficiency requirements for stability, (H_{infty }) robustness, and (H_2) dynamic response, which will hold true if there exist a sliding mode floor matrix (varvec{mathscr {P}}), a symmetric matrix (varvec{mathscr {Q}}), and a matrix (varvec{W}), satisfying:

where (varvec{zeta }_2>0) and (varvec{zeta }_{infty }>0) serve as the weighting parameters.

Perturbation Observer for SFRSMC

It is not feasible to formulate a controller directly using Eq. (26) because the equivalent-based control strategy (u_{eq}) depicted by Eq. (26) includes a composite unknown disturbance term d. It should be noted that if the complex unknown disturbance term d is disregarded and the controller is developed by selecting a sufficiently large switching gain to negate the effect of the complex unknown disturbance term d, this will lead to two issues:

• The controller can only endure a limited number of control inputs due to practical constraints;

• The discontinuity of the control inputs will induce high-frequency jitter in the system, which is detrimental to the system’s stability.

Therefore, the magnitude of the unknown composite disturbance d must be estimated using a disturbance observer and fed back to the controller to actively mitigate the influence of the unknown composite disturbance on the system, thus improving the robustness of the controlled system. Consequently, accurately reconstructing the composite unknown disturbance d is crucial for achieving optimal controller performance.

When the perturbation term varies gradually, it may be assumed that (dot{d}=0) is satisfied, subsequently enabling the standard perturbation observer to be designed as follows:

The location (hat{d}) represents an approximation of the external interference d to the system, while (varvec{L}_o) is the disturbance observer designed to address this.

To enhance the resilience of the system, the error compensation duration (operatorname {sgn}(hat{d}-d)) is implemented, and the disturbance observer might be structured as follows:

From Eqs. (37) and (38), the dynamics equation for the error (e_d=hat{d}-d) related to the perturbed observer can be expressed as:

As the derivatives of the state variables in Eq. (38) cannot be measured in real-time, obtaining these derivatives would amplify noise within the state variables, thereby affecting the observer’s effectiveness. Consequently, a disturbance observer designed with the aid of an intermediary auxiliary variable (z=hat{d}-varvec{L}_ovarvec{X}) is proposed as follows:

The refined formulation for the disturbance observer is then derived by differentiating the intermediary auxiliary variable (z=hat{d}) as follows:

Eq. (41) illustrates that with the assistance of the variable z, the disturbance observer does not need to compute the differentiation of the state variable, thus avoiding noise amplification during the estimation process, making it straightforward to implement in practice.

Proof: To demonstrate the asymptotic convergence of the estimation error (e_{d}) to zero under the condition that the observer gain (varvec{L}> 0), we select the following Lyapunov function:

Taking the derivative with respect to time of Eq. (42), we obtain:

From Eq. (43), it can be observed that when the observer gain matrix (varvec{L}_o> 0), the constructed observer is stable.

In summary, by selecting a positive observer gain matrix (varvec{L}_o), the estimation error (e_{d}) will converge asymptotically to zero, confirming the stability of the designed disturbance observer represented by Eq. (41). This stability characteristic ensures accurate reconstruction of the composite unknown interference d, contributing to the overall robustness and excellent performance of the controller for the HTGS system.

When the system is modeled with high uncertainty and disturbances, the gain of the switching duration should be substantial, which may result in significant jitter. To mitigate jitter, the saturation function (operatorname {sat}(varvec{{*}})) is utilized instead of the symbolic function (operatorname {sgn}(varvec{{*}})) within the controller as outlined in this paper.

where (Delta _b) indicates the thickness of the boundary layer.

The saturation function characterizing Eq. (44) enables the SFRSMC to employ switching control outside the boundary layer and feedback control within the boundary layer. Thus, SFRSMC ensures the system’s control state stabilizes rapidly during management and decreases jitter during sliding mode transitions.