Design of a novel state-feedback strong  $$H_2/H_{infty }$$  sliding-mode controller for a hydraulic turbine governing system

Design technique of the controller

The HTGS mannequin contemplating the management sign uncertainty of the dynamic system in addition to exterior enter disturbances is described by Eq. (17). This paper combines equivalent-based SMC idea and auxiliary feedback-based SMC evaluation to design a SFRSMC in HTGS.

On this paper, we introduce the idea of the sliding floor (varvec{mathscr {S}}) for the HTGS. The sliding floor is related to the system state vector (varvec{mathscr {X}}) and is outlined as follows:

the place (varvec{B}_u^T) represents the transpose of the management enter matrix, and (varvec{mathscr {P}}) is a constructive particular matrix. The selection of the constructive particular matrix (varvec{mathscr {P}}) is important for reaching the mandatory management properties within the HTGS.

By rationally designing the sliding matrix (varvec{mathscr {P}}), we will be certain that the sliding floor (varvec{mathscr {S}}) reveals fascinating conduct, resulting in efficient regulation and stabilization of the HGU. The final word aim is to realize (varvec{mathscr {S}}rightarrow 0) because the system state vector (varvec{mathscr {X}}) converges to the specified state, thereby guaranteeing correct regulation and stability within the HTGS.

For the HTGS, the mathematical expression of the SFRSMC designed on this paper might be described as:

the place (u_{eq}) is the equivalent-based management regulation; (u_n) is the strong management regulation.

In accordance with the equal management precept, and by taking (Delta u = 0), we will get hold of from Eq. (16) and (dot{varvec{mathscr {S}}}=0) the next expressions:

Consequently, the equivalent-based management regulation (u_{eq}) might be decided as follows:

Moreover, based on the design precept of SMC, to make sure that (varvec{S}dot{varvec{S}}le 0), the strong management regulation (u_n) within the controller output (varvec{u}) is outlined as follows:

the place (varepsilon _0) is the switching achieve; (operatorname {sgn}(*)) is the symbolic perform.

Subsequent, the dynamics of the sliding floor (varvec{mathscr {S}}) is analysed by the Lyapunov’s stability theorem, and the Lyapunov perform is chosen as:

The by-product of the sliding floor (varvec{mathscr {S}}) is obtained:

Then, the by-product of the Lyapunov perform (varvec{V}_{S}) outlined by Eq. (23) is obtained:

In accordance with Lyapunov’s stability theorem, when (dot{varvec{V}}_Sle -varepsilon _0|varvec{mathscr {S}}|), when (trightarrow infty), it’s attainable to make (varvec{mathscr {S}}rightarrow 0).

In conclusion, the mathematical expression of the SFRSMC for the HTGS is obtained by combining Eq. (21) and Eq. (22):

On this expression, a mixture of phrases determines the management enter u. The primary half is derived from the equivalent-based management regulation (u_{eq}). This time period ensures that the management regulation steers the system towards the specified sliding floor and tracks the reference state. The second half (u_{n}) incorporates the sliding-mode strong management facet of the SFRSMC technique. It introduces a sliding mode strategy to reinforce robustness towards uncertainties and disturbances within the HTGS. The ultimate management regulation given by Eq. (26) gives a sensible and strong strategy for regulating the HGU and reaching stability and exact management within the HTGS operation.

SMC evaluation of auxiliary feedback-based

The above evaluation exhibits that the design of the symmetric constructive particular matrix (varvec{mathscr {P}}) has a decisive affect on the management efficiency of state suggestions. On this paper, we discuss with? for the design thought of SMC evaluation primarily based on auxiliary suggestions. Primarily based on this concept, we innovatively introduce the blended (H_2/H_{infty }) LMI to design the silding matrix (varvec{mathscr {P}}) in order that the controller has optimum regulation and the most effective robustness efficiency. First, the mathematical expression of the controller is rewritten as:

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

After incorporating the rewritten controller given by Eq. (27) into the state-space equations described by Eq. (17), the closed-loop system is obtained as follows:

the place (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 dynamical system described by Eq. (28), two units of management efficiency output vectors are added, one for the (H_2) management efficiency output vector and one for the (H_{infty }) management efficiency output vector, to acquire the next augmented dynamical system:

the place (varvec{Z}_{2} in mathbb {R}^{9 occasions 1}), (varvec{Z}_{infty } in mathbb {R}^{9 occasions 1}) are the (H_2) and (H_{infty }) management efficiency output vectors, respectively.

On this augmented system, we’ve got two management efficiency output vectors (varvec{Z}_{2}) and (varvec{Z}_{infty }), that are influenced by the matrix (varvec{C}_{infty }), (varvec{D}_{infty 1}), (varvec{D}_{infty 2}), (varvec{C}_2) and (varvec{D}_{2}), respectively. The augmented dynamical system, described by Eq. (29), permits us to research and design the controller primarily based on completely different management efficiency standards, particularly, (H_2) and (H_{infty }) management efficiency, offering a complete framework for controlling and optimizing the HTGS regarding these efficiency targets. On this paper, the weighting matrix (varvec{C}_{infty }), (varvec{D}_{infty 1}), (varvec{D}_{infty 2}), (varvec{C}_2), and (varvec{D}_{2}) in Eq. (29) is outlined 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}) are the systematic coefficient matrices of the suitable dimensions; it must be famous that (varvec{Q}_{infty }), (varvec{R}_{infty }), (varvec{R}_ {infty }), (varvec{Q}_{2}), and (varvec{R}_{2}) are outlined on this paper as:

the place ({ q}_i ge)0 ((i=1), (cdots), 8) and ({ r}) are the weighting components.

In accordance with the sliding floor (varvec{mathscr {S}} = varvec{B}_u^Tvarvec{mathscr {P}}varvec{X}) outlined on this paper, the matrix (varvec{mathscr {P}}) primarily serves to prepare and fuse the state suggestions info of the managed object. Primarily based on this concept, this paper defines the connection between the state suggestions matrix (varvec{Okay}) and the constructive particular matrix (varvec{mathscr {P}}) in (28) as (varvec{Okay} = varvec{W}varvec{mathscr {P}}). The matrix (varvec{mathscr {P}}) is solved by designing an appropriate matrix (varvec{Okay}) such that the managed object described by Eq. (28) has blended (H_2/H_{infty }) management efficiency. The blended (H_2/H_{infty }) management efficiency requires that the closed-loop system meets the next efficiency:

1) (H_{infty }) robustess efficiency, the closed-loop switch perform (widetilde{varvec{G}}_{z_{infty }d^{*}}(s)) from (d^{*}) to (varvec{Z}_{infty }) satisfies:

The (H_{infty }) efficiency by a suggestions achieve matrix (varvec{Okay}_{infty } = varvec{W}_{infty }varvec{mathscr {P}}_{infty }) is achieved if and provided that the matrices (varvec{W}_{infty }) and (varvec{mathscr {P}}_{infty }) exist and the next LMI is happy:

the place (langle * rangle _s=(*)+(*)^T).

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

The (H_{infty }) efficiency by a suggestions achieve matrix (varvec{Okay}_2 = varvec{W}_2varvec{mathscr {P}}_2) is achieved if and provided that the matrices (varvec{W}_{2}), (varvec{mathscr {P}}_{2}), and (varvec{mathscr {Q}}) exist and the next LMI is happy:

the place (textbf{tr}(varvec{N})) is the hint of matrix (varvec{N}).

For comfort, allow us to set (mathscr {P}_{infty }=varvec{mathscr {P}}_2 triangleq varvec{mathscr {P}}) and (varvec{W}_{infty }=varvec{W}_2 triangleq varvec{W}). The closed-loop suggestions achieve matrix by (varvec{Okay}=varvec{W}varvec{mathscr {P}}) that concurrently satisfies the efficiency required by stability, (H_{infty }) robustness efficiency, and (H_2) dynamic response, which has an answer if there exist sliding mode floor matrix (varvec{mathscr {P}}), a symmetric matrice (varvec{mathscr {Q}}) and a matrix (varvec{W}), satisfying:

the place (varvec{zeta }_2>0) and (varvec{zeta }_{infty }>0) are the weighting components.

Perturbation observer for SFRSMC

It’s not attainable to assemble a controller instantly utilizing Eq. (26) as a result of the equivalent-based management regulation (u_{eq}) described by Eq. (26) has a composite unknown disturbance time period d. It must be famous that if the advanced unknown disturbance time period d is eliminated and the controller is constructed by deciding on a big sufficient switching achieve to cancel the impact of the advanced unknown disturbance time period d, which results in two issues:

• The controller can solely stand up to a restricted variety of management inputs attributable to sensible constraints;

• The discontinuity of the management inputs will trigger high-frequency jitter within the system, which isn’t conducive to the steadiness of the system.

Subsequently, the magnitude of the unknown composite disturbance d must be estimated by the disturbance observer and suggestions to the controller to actively scale back the affect of the unknown composite disturbance on the system, thus enhancing the robustness of the managed system. Subsequently, correct reconstruction of the composite unknown disturbance d is the important thing to reaching the superb efficiency of the controller.

When the perturbation time period adjustments slowly, it may be assumed that (dot{d}=0) is happy, then the standard perturbation observer might be designed as:

the place (hat{d}) is an estimate of the exterior disturbance d to the system, and (varvec{L}_o) is the disturbance observer achieve.

To boost the robustness of the system, the error compensation time period (operatorname {sgn}(hat{d}-d)) is launched and the perturbation observer might be designed as:

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

Because the derivatives of the state variables in Eq. (38) can’t be measured instantly, discovering the derivatives would amplify the noise of the state variables and have an effect on the effectiveness of the observer. Subsequently, a perturbation observer designed with the assistance of intermediate auxiliary variable (z=hat{d}-varvec{L}_ovarvec{X}) is proposed as:

The improved expression for the perturbation observer is then obtained by deriving the intermediate auxiliary variable (z=hat{d}) as:

Eq. (41) exhibits that with the assistance of the intermediate variable z, the perturbation observer doesn’t must calculate the state variable differentiation and doesn’t trigger noise amplification throughout the statement course of, which is simple to implement in engineering.

Proof: To show the asymptotic convergence of the statement error (e_{d}) to 0 when the observer achieve (varvec{L}> 0), we select the next Lyapunov perform:

Taking the time by-product of Eq. (42), we’ve got:

From Eq. (43), it may be seen that when the observer beneficial properties matrix (varvec{L}_o> 0), the designed observer is steady.

In abstract, by deciding on a constructive observer achieve matrix (varvec{L}_o), the statement error (e_{d}) will converge asymptotically to 0, confirming the steadiness of the designed perturbation observer described by Eq. (41). This stability property ensures correct reconstruction of the composite unknown disturbance d, contributing to the general robustness and wonderful efficiency of the controller for the HTGS system.

When the system is modeled with excessive uncertainty and disturbances, the achieve of the switching time period must be massive, and this causes massive jitter. To stop jitter, the saturation perform (operatorname {sat}(varvec{{*}})) is used as a substitute of the symbolic perform (operatorname {sgn}(varvec{{*}})) within the controller on this paper.

the place (Delta _b) is the boundary layer thickness.

The saturation perform described by Eq. (44) permits the SFRSMC to make use of switching management exterior the boundary layer and suggestions management contained in the boundary layer. Subsequently SFRSMC makes the management state of the system stabilize shortly throughout management and reduces jitter throughout sliding mode switching.