An Improved Second-Order Sliding Mode Control for an Interception Guidance System without Angular Velocity Measurement

: It is well-known that the successful implementation of current sliding-mode guidance laws relies on two key points: one is the prior knowledge of the upper bound of the target’s maneuver, and the other is the accurate measurement of motion information. The truth maneuver is usually difﬁcult to acquire and can only be roughly estimated. Measurements are often affected by noise and physical limitations of the onboard sensor. This paper presents an improved second-order sliding-mode guidance law that can handle these two problems simultaneously while addressing the chattering phenomenon, which inherently exists in the sliding mode-based controller. We achieve this by using the estimation from an extended state observer (ESO) and tracking differentiator (TD). In light of the employed ESO, which is utilized to give an estimation of the external disturbance, prior knowledge of the target’s maneuver is not required. Most interestingly, we show that the motion information can be estimated efﬁciently from the ESO in real time. A TD ﬁlter can be further embedded to remove unwanted noise from the detected signal. A series of simulations has been conducted, clearly demonstrating that the estimation accuracy of the ESO is sufﬁcient for the guidance law to implement accurate interception. The TD ﬁlter not only removes noise but also avoids the phase-loss problems associated with conventional ﬁlters. Moreover, the chattering phenomenon is completely eliminated in the control channel.


Introduction
With the rapid development of weapons, targets of airspace intrusions may deploy countermeasures, such as attitude controllers, thrusters, or interference devices, such as infrared decoys and balloons. Although many powerful controllers and detectors have been leveraged [1][2][3], it is still significantly difficult to intercept the target with countermeasures. To implement an accurate intercept against a moving target with countermeasures, the missiles should have proper sensors and employ real-time guidance laws.
Guidance laws are key to the successful implementation of interceptions. The speed and maneuverability of targets are increasing, leading to the interception systems becoming progressively complex, characterized by nonlinearity, time-varying, and model uncertainty; moreover, traditional guidance laws have been difficult to apply. Therefore, many scholars have attempted to apply powerful nonlinear control methods to design guidance laws, aiming to improve the success rate and reduce the sensitivity of missiles to maneuvering targets, as well as reduce the fuel requirements of missiles. Biased proportional navigation guidance [4] methods incorporate the appropriate bias components to align the guidance law to meet the requirements of a specific angle of impact, thus increasing the effectiveness of the impact. In addition, there are some optimal guidance laws [5,6], especially for highspeed maneuvering targets. These methods incorporate time-varying control constraints into the objective function to reduce energy consumption or apply integral commands, to reduce missile sensitivity to target maneuvers. In real-world applications, the above methods have achieved good results but they all require an estimation of the time-to-go. However, the estimation of time-to-go allows the controller to only obtain suboptimal solutions. Moreover, the computational cost of the optimal control law is an inherent difficulty.
Sliding-mode guidance laws do not require an estimation of time-to-go. In addition, the sliding mode control possesses unique characteristics, such as its invariance to parameter perturbations and its robustness against external disturbances; therefore, it is widely exploited in the navigation and guidance field [7]. Sliding-mode guidance laws for nonlinear systems, such as [8], achieve interception against high-speed maneuvering targets at specific angles of impact. To make the sliding-mode guidance law perform as expected in practical applications, the upper bound of the target's maneuver should be estimated in the first place, and then the gain of the sign function term should be adjusted. Too high of a gain will cause chattering, while if it is too small, the target cannot be intercepted effectively. In light of this, detecting and identifying the unknown target's maneuver is essential for a successful interception. References [9,10] introduced an observer to estimate external disturbances, and the obtained estimation was utilized to develop guiding laws with feedforward compensation. In [7,11,12], a disturbance observer was employed to estimate the upper bound of the target's maneuver. However, few methods take into account the effects of noise. Different from the above methods, we embedded a TD filter into the observer system to remove unwanted noise from the detected signal.
Terminal sliding mode (TSM) [13] and fast terminal sliding mode (FTSM) [14] were proposed to tackle the problem of finite-time convergence. Finite-time convergence is of considerable importance as it signifies that the system state can reach the desired value within a finite duration. To overcome the singularity problem that inherently exists in TSM and FTSM, the non-singular terminal sliding mode (NTSM) [15] guidance law utilizes a nonlinear engagement dynamics sliding mode surface such that when the control reaches saturation, the sliding variable can still converge to zero in finite time. To suppress the high-frequency chattering that appears in the control channels, NTSM utilizes a continuous saturation function. However, replacing the symbolic function with a saturation function results in a dynamic error that will cause the control accuracy to be lower than the desired accuracy. Even with the saturation function, the chattering phenomenon is still serious in the face of high-maneuvering targets. Our method was developed from the NTSM scheme, adopting a new strategy to completely eliminate the chattering phenomenon without sacrificing system performance.
On the other hand, against high-speed targets with countermeasures, detection must be done early enough to leave enough time for terminal guidance. Passive infrared provides sufficient accuracy for detection when the distance is less than 20 km [16]; however, it is sensitive to infrared radiation countermeasures. Another solution is laser radar. Infrared radiation countermeasures do not affect the detection of the laser radar and, owing to its shorter wavelengths with the same aperture, laser radar can provide a higher resolution range [16]. The extension of the sensing range provides more available time for terminal guidance. The onboard sensor not only provides the range measurement but also infers information about the motion knowledge of the target, such as angular velocity. Yet, angular velocity detection requires a very high angular resolution, which is a great challenge for existing sensors. One possible solution is to develop a guidance law that takes into account the limitations of the sensor. We draw inspiration from Reference [17], which employed a linear filter to estimate higher-order derivatives of the angle of attack and sideslip. In our work, we employed ESO to estimate the first-order derivative of the line of sight (LOS) angle, which was directly utilized in constructing the guidance law.
Based on the achievements of TSM, FTSM, and NTSM, we developed an improved continuous second-order sliding-mode guidance law by exploring the complementary benefits of the ESO and TD. The new guidance law has the following characteristics, which can potentially improve interception performance.

1.
The proposed guidance law does not require angular velocity information since ESO has its potential role as part of a sensor, whose output is the estimation of the angular velocity. Most existing observers have limited applications, primarily focused on observing external uncertainty disturbances. Therefore, our work extends the usefulness of the observer.

2.
To completely eliminate the chattering phenomenon without sacrificing the convergence accuracy and convergence speed of the system, an adaptive gain of the discontinuous control is introduced. As a result, the minimal possible value of the discontinuity magnitude was found for the current state. We also introduce an active disturbance rejection observer to estimate external disturbances; the estimation was used to derive guiding laws with feedforward compensation, which further improved the robustness to external disturbances.

3.
A TD filter was employed to cope with the noisy problem and avoid the phase-loss phenomenon associated with conventional filters.

Relative Motion of Interception Process
The relative motion between a missile and a target is derived from the terminal guidance law. Figure 1 depicts one possible interception that is ballistic in the longitudinal plane. The positions of the missile and target are described in the Cartesian coordinate system, where the subscripts M and T represent the physical variables belonging to the missile and target, respectively. R is the relative range between the missile and the target, and q is the LOS angle. θ M and θ T are the angles between the velocity and the horizontal baseline. v M and v T are the velocities of the missile and the target, respectively. a M and a T are the accelerations of the missile and the target, respectively. η M = q − θ M and η T = q − θ T are the angles between the velocity and the LOS. According to Figure 1, the formulas of motion for this problem are as follows: whereṘ is the relative velocity andq is the LOS angle velocity; note that it is difficult to obtain an accurate measurement ofq using the sensor. During the guidance period, which is before R = 0, q satisfiesq T q = 0 orq ⊥ q. According to the parallel approach principle, the LOS angular velocity converges to 0, which means that the interception is successfully achieved. At this point, we can obtain: The subscript t indicates the values corresponding to the end of the interception. The collision angle constraint specifically refers to the angle α d between v M t and v M t at the moment of interception. Based on the definitions of α d , q, and Equation (2), the following lemma is obtained: ). For a guidance problem with a collision angle constraint, the desired collision angle α d always has a unique and definite LOS angle q d corresponding to it.
Thus, the collision angle control problem can be converted to the LOS angle control problem. Ultimately, the control targets of this paper translate into designing a controller that achieves q → q * ,q → 0.

Dynamics for LOS Angle Tracking Error
In the previous subsection, the equations of motion were derived based on the interception geometry. The control objectives were defined as follows: designing a controller to drive the LOS angular rate to converge to zero and ensuring that the LOS angle also converges to the desired value. In the control, it is customary to control variables that converge to the relative origin, so we define the LOS angular error and derive the form of the normal acceleration by calculating its time derivative. First, we find the time derivative for (1).R = Rq 2 + a Tr − a Mr , whereR is the relative acceleration between the missile and target,q is the LOS angular acceleration. a Tr and a Mr are components of the target acceleration and missile acceleration along the LOS, defined as: a Mr := a M sin(q − θ M ), a Tr := a T sin(q − θ T ). a Tq , and a Mq are the components of the target acceleration and missile acceleration that are orthogonal to LOS, respectively, defined as: a Mq := a M cos(q − θ M ), a Tq := a T cos(q − θ T ). Normally, in the guidance phase, the axial acceleration of the missile is an uncontrollable variable, but the normal acceleration a Mq of the missile can be controlled by adjusting the rudder deflection angle. Therefore, the relative second-order relationship between the input signal a Mq and the LOS angle q is used to design a guidance law. Then we define the LOS angular tracking error: where q d denotes the desired LOS angular. The derivative of (4) givesė =q. The dynamics equation for the LOS angle error is obtained by bringing in (3) We can observe that when q − θ M = ± π 2 , a Mq = 0 m/s 2 , the controller cannot exert control over the system and, therefore, treats this state point as a singularity. To avoid the singularity phenomenon, (5) is rewritten as follows: where a M is the virtual control input, and d is seen as the general uncertainty term containing the random external disturbance a Tq . The use of the virtual control input a M and the definition of the general uncertainty term make the system fundamentally immune to the existence of singularities. Therefore, the control objective changes to design a virtual control input a M , such that e → 0,q → 0.

Assumption 1.
For missiles and targets, their accelerations are assumed to be bounded due to the presence of physical constraints, and they satisfy the following condition: |a T | ≤ a max T , |a M | ≤ a max M . Therefore it is also possible to derive |d| ≤ d max , where a max T is the unknown upper bound target acceleration.

Guidance Law Design
With the error dynamic Equation (6), we illustrate the control system in Figure 2. For each time step, ESO provides the estimated state and uncertainty for the continuous second-order sliding mode (CSSM) controller. The CSSM controller contains two main components, namely the TSM sliding surface and an adaptive gain. Next, we introduce these components and derive the proposed CSSM guidance law.

ESO for the General Uncertainty Term
The ESO will be used to make real-time observations of the uncertainty term. We consider the following system of arbitrary order: where θ(t) and u(t) denote the system state and the control input, respectively. d(t) is seen as the uncertainty term. We assume that the state is nth-order derivable and satisfies ḋ < L, where L is a scalar. For (7), the ESO can be utilized to estimate d(t) in real time, and the observer is represented by a system of differential equations [18]: ] is the estimation of the state and the derivatives of each order of the state,d is the estimation of the uncertainty term, r is the input signal, ∂ i (i = 1, 2, 3, . . . , n + 1) are the ESO parameters to be designed. The stability of the ESO observer will be proved in Appendix A.

Remark 1.
One can see that the ESO can estimate the uncertainty term as well as the higher-order derivatives of the state. Differentiating ourselves from previous works that solely utilized observers for estimating external disturbances, we are the first to investigate its potential role as part of a sensor, whose output serves as the estimation of the higher-order derivatives of the state, such as the angular velocity. Using the estimated LOS angular rate from the ESO to calculate the guidance law, we can address the specific limitation of the sensor. In this way, this not only makes full use of the observer but also reduces the accuracy requirement of the sensor.

Non-Singular Terminal Sliding Mode Control Strategy
The newly developed non-singular terminal sliding mode (NTSM) control strategy [19] eliminates the singular phenomenon while ensuring the system converges to the origin in finite time. Considering a second-order nonlinear system: where x = [x 1 , x 2 ] ∈ R n represents the system state, u ∈ R m is the control input, F(·), B(·) are continuous functions, and D(·) represents the external disturbance, which is assumed to satisfy |D(x)| ≤ L max , where L max is the upper bound of the disturbance. In the NTSM scheme, the sliding surface is designed in the following form: To stabilize the system (9), the controller is designed as follows: where µ, σ are the design parameters. Then the system (9) will converge to the origin in finite time.

Remark 2.
To inhibit the chattering phenomenon, one can replace the discontinuous sign function with continuous saturation or the sigmoid function in (11). However, this can lead to some problems. Firstly, replacing the sign function with a saturation function results in a dynamic error that will cause the control accuracy to be lower than the desired accuracy [20]. Secondly, even with the saturation function, the chattering phenomenon is still serious in the face of high-maneuvering targets. This serves as motivation for the development of a sliding mode control scheme that eliminates chattering without sacrificing performance.

Continuous Second-Order Sliding-Mode Guidance Law
The design of the sliding mode controller consists of two main aspects: 1.
The design of the sliding surface s; 2.
The construction of the control law a M .
The choice of the sliding surface is related to the dynamic characteristics of the system. The control law ensures that the sliding mode is reachable and finite-time stable. The super-twisting algorithm (STA) provides a new perspective to eliminate the chattering phenomenon. In the STA, an adaptive gain function is introduced. As a result, the minimal possible value of the discontinuity magnitude is found for the current state to reduce the amplitude of chattering. We draw on its idea to improve the traditional NTSM control. We first introduce STA [21] and then design a continuous second-order sliding guidance law.
We consider a second-order nonlinear dynamical system: where x, y are states; we assume d dt |φ| ≤ L, and the control input is defined as: where k(t) is a state-dependent gain function that determines the magnitude of the chattering amplitude and follows the adaptive law: This adaptive law aims to reduce the magnitude of the gain of the discontinuous function term sgn(x) to suppress chattering, and k(t) will stop updating when the system enters a sliding mode.
Reference [21] indicates that, for the system in (12), if using the control input (13) with gain, updated according to (14), then the state will converge to 0 in finite time. The proof is seen in Appendix B.
The TSM surface [13] ensures superior dynamic performance, including excellent accuracy and high convergence speed. In this section, in order to realize the fast and accurate interception, a TSM surface is used, and the sliding surface contains the LOS angle error and the first-order derivative of the LOS angle error to satisfy the collision angle constraint. The surface is designed as follows: where κ > 0, 0 < α < 1 are parameters to be designed. According to SAT, let x = s, and construct a guidance law according to (13), given by: whereq is the output of the ESO, and p 1 > 0, p 2 > 0,γ > 2 are parameters to be designed.

Remark 3.
The gain functions k 1 (e), k 2 (s) of the discontinuous term sgn(·) are state-dependent continuous functions and are adaptively adjusted according to the state, which ensures the sliding mode is reachable while minimizing the gain value and suppressing the high-frequency chattering caused by the discontinuous function.

Stability Analysis
Theorem 1. For the LOS angular error dynamic in (6), with Assumption 1, using the TSM surface in (15), the guidance law in (16), and the ESO in (7), the conditions for the completion of the interception task e → 0,q → 0 can be achieved in finite time.
Proof of Theorem 1. The derivative of (15) yields: Bringing Equation (16) into the above equation, we obtain: Assuming that the ESO can accurately estimate the uncertainty term and LOS angular velocity, Equation (18) can be rewritten as: We introduce new variables: Taking the time derivative of (20) yields:ρ Since p 1 > 0, p 2 > 0, γ > 2, κ > 0, 0 < α < 1, it can be proven that Ξ is a Hurwitz matrix. Thus, for any positive definite symmetric matrix P = P > 0, there exists a positive definite symmetric matrix Q = Q > 0 that satisfies the following condition: We define a candidate Lyapunov function: Finding the time derivative for V ρ , and bringing (21) and (22), we obtain: According to the definition of ρ = | 1 | 1−1/γ sgn( 1 ) 2 T , and γ > 2, it can be inferred that | 1 | γ−1 γ ≤ ρ 2 , where the symbol · 2 represents the second norm of the vector; thus, (24) can be expressed as: where λ min (·), λ max (·) denote the minimum eigenvalues of the matrix and maximum eigenvalues of the matrix, respectively. Since Using the finite-time bounded function approach and Lyapunov stability criteria, one can estimate the finite time t f : when |ė| = 0,t > t f , we can obtain: It can be seen that when t > t f , if s = 0, then |ë| = 0. This indicates |ė| = 0 is not an attractor and does not affect the reachability of the sliding surface. Once the system reaches the sliding surface: e + κ|ė| α sgn(ė) = 0, It can be proven that system (29) is finite-time stable. First, we define the Lyapunov candidate function: Then, taking the time derivative of V e yields: since κ > 0, 0 < α < 1, we can obtainV e ≤ 0. In summary, (29) is finite-time stable, and it follows that (17) is finite-time stable.

Simulation Analysis
To verify the effectiveness of the proposed guidance law, detailed numerical simulations are implemented.

Simulation 1
The improvement of the guidance law by using the ESO and tracking differentiator (TD) filter is analyzed in simulation 1.
Assume that the target normal acceleration is in the form of a periodic square wave, maximum normal acceleration 40 m/s 2 , and minimum normal acceleration −20 m/s 2 . Suppose the velocity of the missile is 400 m/s and the available normal acceleration of the missile is ±200 m/s 2 . The velocity of the target is 200 m/s, the initial position of the missile is (0, 0)(m), and the target is (10000, 10000) In this section, the performance of the ESO is compared with that of the disturbance observer (DO) [22], and the form of DO is given below. First, the system needs to be rebuilt, let ξ = Rq, and thenξ =Ṙq + Rq. Substituting into Equation (3) The reconstruction of acceleration information is u = −Ṙq − a M . The final system with simplified representation is obtained asξ = u + d. For the system mentioned above, assuming d [2] < L, the form of the DO is given below: where Eventually, the output of the DO observer v 0 is the estimation of ξ, v 1 is the estimation of d, v 2 is the estimation ofḋ, and the parameters of DO are designed as follows: λ 0 = 1.1, In order to demonstrate the improvement of estimation compensation in the performance of the guidance law, we also introduce a guidance law without estimation compensation as a baseline for comparison; the formulation of the no-observation-compensate (NOC) guidance law is as follows: Compared to (16), (33) omits the compensation term Rd, while the remaining parameters are the same as those in the CSSM guidance law listed in Table 1.  Figure 3a shows the trajectory of the missile and the target, respectively. It can be observed that the ballistics of the missiles are smooth. As shown in Figure 3b, the acceleration commands reach the saturation values in 0-5 s and no chattering occurs. Figure 3c,d show the relative velocity and relative distance, respectively. Figure 3e,f show that LOS angular rates converge to 0 in finite time and LOS angles converge to the corresponding desired values, indicating that the missile intercepts the target with desired collision angle. Figure 3g shows that the sliding variables achieve 0 in finite time, the curve is continuous, and no chattering occurs. The results mentioned above demonstrate that the guidance law constructed using the LOS angular rate, estimated by ESO, can also meet the guidance performance requirements. Figure suih compares the performance between ESO and DO. The ESO reduces the initial peak by about 50% and converges to the true value more quickly. In particular, in the case involving abrupt changes in the true value, the ESO can still perform accurate tracking with no overshoot and no oscillation in the tracking process, and respond quickly, with significantly better performance than the DO observer.
In Figure 3i, design 1 represents the guidance law with the ESO estimation compensation, and design 2 represents the guidance law without estimation compensation. It can be seen that both sliding variables converge to near 0, and the performance improvement of the observer is very obvious, with the sliding variable achieving the steady state more quickly and with higher control accuracy. In fact, we also find that as the target maneuverability increases, the performance improvement from the observation compensation becomes progressively more apparent. Overall, compared with the DO guidance law and NOC guidance, the proposed CSSM guidance law achieves better performance in terms of the flight time, miss distance, and angle tracking error (see Table 2).

Simulation 2
In simulation 2, we consider the problem of noise interference appearing in the LOS angle measurement channel. Note that few of the existing related works consider the effect of noise on the observer; in practice, this is a problem that has to be addressed. To demonstrate that our method can cope with arbitrary forms of target maneuvers, the normal acceleration of the targets in simulation 2 and simulation 3 are set as a T = 60 sin(0.3t) m/s 2 . The magnitude of the noise is shown in Figure 4b. In the presence of noise, as shown in Figure 4a, the missile misses the target. Figure 4c shows that noise makes the observer generate a severe hysteresis appearance, resulting in an inaccurate estimation of the uncertainty term. To address this problem, we introduce a TD filter [23] to remove unwanted noise from the detected signal. Compared with the classical filter, TD replaces the inertia segment of the classical filter with a second-order form to eliminate or reduce the amplification effect of noise. The second-order filter has a lower noise amplification gain than the classical filter, and TD can also reduce the noise of the differential prediction by increasing the sampling frequency, which is not possible with the classical filter and is, therefore, suitable for missile interceptor systems where measurement noise exists. The form of TD [23] is as follows: where w * indicates the noise-contaminated LOS angle measurement signal. w 1 , w 2 are the states of the TD, representing the filtered values. Since filtering always results in the phase loss, this method compensates for phase loss by utilizing the differential signal from the TD output and applying appropriate forecast correction;ŵ is variable after the forecast correction; χ is the design parameter that regulates the tracking speed of TD, and its value in the simulation is chosen as χ = 50; h is the simulation time step, whose value is set as h = 0.01 s.  As seen in Figure 4c, the signal is filtered and fed to the observer, which allows the observer to capture the characteristics of the uncertain signal more accurately; however, a burr exists in the observations. As shown in Figure 4a, even in the presence of measurement noise, the missile can still hit the target with a straight trajectory. In summary, we can conclude that TD can efficiently eliminate the effect of noise on the guidance system and enable the missile to complete the interception mission.
It should be pointed out that with both the ESO and TD filter methods, the parameters have a huge impact on their performance. Sophisticated tuning is required to achieve optimal performance. This is also the main restriction limiting the application of this method.

Simulation 3
To further demonstrate the effectiveness of the proposed algorithm in eliminating the chattering phenomenon, the non-singular terminal sliding mode (NTSM) algorithm will be utilized as a comparison. For the system in (6), according to (10) and (11), an NTSM-based guidance law is designed as follows: where α = 0.71, κ = 1.4 are the same as the setting in the CSSM. The gain of sgn(s N ) is set at K = 400. Note that the selection of parameter K is related to the upper bound of the target acceleration, which is usually required to be roughly estimated in advance. As a comparison, in our method (16), prior information on the target acceleration is not required because it is provided by the ESO in real time. No excessive K fundamentally addresses the chattering phenomenon, while the estimated value will be used to construct additional compensation terms to ensure sufficient robustness against external disturbances. Figure 5a,c show that two guidance laws based on NTSM and CSSM successfully intercept the target. As depicted in Figure 5e, both NTSM and CSSM guidance laws satisfied the terminal collision angle constraint. From Figure 5d,f, it can be seen that the LOS angular rate and sliding variable converge to 0 in finite time. However, as shown in Figure 5b, the NTSM guidance law exhibits a significant chattering phenomenon in the control channel, which is because, as the target maneuverability increases, the value of K needs to be large enough to meet the system's robustness requirements. Unfortunately, the amplitude of chattering is proportional to K. From this point of view, our method benefits a lot from the proposed adaptive gain function since it can automatically find a minimum possible value of the discontinuity magnitude for the current state so as to significantly reduce the amplitude of chattering. Overall, compared with the NTSM guidance law, the proposed CSSM guidance law completely eliminates the chattering phenomenon and achieves a higher control accuracy (see Table 3).

Conclusions
Inspired by active disturbance rejection control techniques, we have designed an intercept guidance law that is insensitive to uncertainties and noise in the external environment. The proof of finite-time stability is given in detail by using the finite-time bounded method and Lyapunov functions. The ESO estimates the uncertainty term containing the target maneuver, and the estimation is used to derive the sliding-mode guidance law with feedforward compensation. We explored the potential role of the ESO as part of a sensor, i.e., the output of the ESO can be used to construct a guidance law that compensates for the limitation that the sensor on board cannot detect the angular velocity.
Simulation results demonstrate that accurate interception can be achieved under the guidance law using the LOS angular velocity estimated by ESO. The control objective is completed in finite time due to the use of a finite-time convergence algorithm, STA, and chattering is completely eliminated. In addition, the simulations also demonstrate the improvement of the observer on the guidance performance in terms of the mission completion time and control accuracy. The simulation results show that the TD filter eliminates the noise effectively so that the missile can successfully complete the interception mission and avoid the phase-loss phenomenon.  Data Availability Statement: Unavailable due to privacy.

Conflicts of Interest:
The authors declare no conflict of interest. Lemma A1 ( [18]). For a Hurwitz matrix E and any positive definite symmetric matrix P, there exists a positive definite symmetric matrix Q, satisfying the following condition: E n P + PE + Q = 0.
Next, we define a Lyapunov candidate function, and then take the time derivative of (A5): where λ min (Q) represents the minimum eigenvalue of the matrix Q. Once ζ satisfies the condition: the inequality condition yieldsV E ≤ 0, which means that the observation error of the observer converges to 0. From this, it is also clear that the choice of parameters ∂ i (i = 1, 2, 3, . . . , n + 1) and ε influences the stability and final performance of the ESO.

Appendix B
In the sliding mode arrival state, when y ≡ 0, the equation can be expressed as 0 = φ(t) − k(t) sgn(x(t)). At this point, sgn(x(t)) should be replaced by an equivalent form, denoted by ϑ: The equivalent function Θ[sgn(x(t)) ∈ (−1, 1) can be regarded as an equivalent substitution, in the sense of the mean, for the discontinuous function sgn(x(t)). Defining the Lyapunov candidate function, V ϑ := 1 2 ϑ 2 , the derivative of V ϑ yields the following: By the assumption that φ(t) is derivable and bounded, taking the formula (A8) yields: We bringk(t) = Λk(t) sgn(ϑ(t)) into the above equation: wherek,φ are the upper bounds of the functions k(t), φ(t), respectively. When Λ > L φ is satisfied, then ℵ = L − Λφ < 0, then we can obtain: In summary, algorithm (13) is finite-time convergence.