Enhanced Adaptive Super-Twisting Current-Constrained Control of PMSM Based on Generalized Proportional-Integral Observer

1  Introduction

The permanent magnet synchronous motor (PMSM) has been found to be a highly effective solution for servo systems, owing to its high power density, excellent dynamic performance, and control flexibility [1]. Conventionally, the control of PMSM speed is implemented via a cascade structure consisting of an outer speed loop and an inner current loop. This configuration, historically constrained by hardware limitations, operated the two loops at different rates, resulting in a sequential delay. Such a configuration often complicates parameter tuning and limits the dynamic response. In light of contemporary advancements in computational resources, the execution of both the current and speed loops can be conducted in parallel. This configuration facilitates a non-cascade control architecture, which, in turn, simplifies system design and enhances performance.

The non-cascade structure has been shown to simplify design and reduce the number of adjustable parameters in comparison with its cascade counterpart, thus rendering it a prominent research focus. For instance, reference [2] presents a control method that integrates generalized predictive control with a nonlinear disturbance observer within this non-cascade framework. This approach is designed to enhance the dynamic performance of PMSM systems and streamline parameter tuning. Meanwhile, reference [3] investigates a dual three-phase PMSM system subject to time-varying disturbances and adopts a non-cascade hierarchical control architecture to streamline controller design. Furthermore, reference [4] proposes a fast non-singular terminal sliding-mode speed control strategy under a similar non-cascade structure to overcome the adaptability limitations of conventional PI controllers.

However, in the non-cascade structure, the merging of the current and speed loops changes the q-axis current from an output to a state variable. Consequently, conventional PID controllers are no longer capable of effectively constraining it, which may result in damaging transient overcurrents. Whilst the selection of more conservative (lower-gain) parameters has the potential to mitigate the risk of overcurrent, it is inevitable that this will result in the degradation of the system’s dynamic performance. In order to address transient overcurrent in non-cascade PMSM drives, a range of state-constrained control strategies have been proposed [5]. For instance, Reference [6] proposed a reference trajectory modulation method for nonlinear systems with unknown parameters, which enforces constraints by using the reference signal as a closed-loop input. Nevertheless, ensuring constraints over an infinite horizon remains practically challenging due to overly restrictive requirements on the reference signal class. Alternatively, the invariant set method guarantees that states remain within prescribed limits and can address both input saturation and state constraints [7]. Nevertheless, the computational intricacy inherent in ascertaining the invariant set frequently restricts its applicability to particular initial conditions. Another common approach is model predictive control (MPC), which has been shown to handle constraints by design [8]. Despite its extensive utilisation, MPC necessitates a precise model, exhibits sensitivity to disturbances, and demands considerable computational resources, thereby impeding its practical implementation. To summarise the aforementioned analysis, the following conditions must be satisfied to resolve the overcurrent issue under non-cascade PMSM control: The primary objective is to constrain the current within a limited range. The second objective is to construct an easily implementable control system. The third objective is to ensure that the dynamic steady state performance is satisfactory. In this paper, novel constrained controllers are designed with two objectives in mind. Firstly, these controllers are intended to avoid overly complex constraints, such as the reference track line modulation method. Secondly, they are designed to avoid limiting the initial values, for example, using an invariant set, or requiring an exact system model, for example, using MPC.

In order to address the current constraint in non-cascade PMSM drives, this paper proposes a controller that incorporates a current-limiting penalty mechanism into the adaptive gain design. Specifically, a penalty function is constructed based on the current limit, which approaches infinity as the q-axis current nears the boundary, thereby effectively suppressing transient overcurrent. However, traditional PI control is deficient in terms of robustness against parameter variations and external disturbances. Among the various robust control alternatives, Sliding-mode control (SMC) has been extensively adopted in motor drives, robotics, aerospace, and power electronics due to its robust stability, rapid response, and structural simplicity. However, the practical implementation of SMC is constrained by factors such as finite control bandwidth and discrete sampling, which induce high-frequency chattering. This chattering has the potential to induce substantial torque ripple, thereby compromising system stability. Consequently, the suppression of chattering is imperative for enhancing the steady-state performance of SMC systems [9,10].

In order to mitigate the impact of chattering vibration in the sliding mode control, literature [11] proposes the adoption of fuzzy switching gain adjustment, a methodology that eliminates the interference term through the effective estimation of the switching gain. However, this approach can compromise the robustness of the control system. Literature [12] proposes a finite-time sliding mode control, which aims to suppress the chattering effect associated with conventional SMC within a finite period of time. However, the method requires high mathematical accuracy and is difficult to apply to practical systems. Literature [13] adopts a second-order super-twisting algorithm (STA), which ensures the strong robustness of the system while eliminating the chattering vibration, and thus has become a research hotspot in PMSM control. As described in [14], a fast super-twisting non-singular fast terminal sliding mode controller has been developed. This controller has been shown to attenuate system singularity and chattering phenomenon, improve the switching convergence rate, and exhibit a faster convergence rate in comparison to traditional second-order sliding modes. Literature [15] proposes a novel generalized adaptive super-twisting algorithm for the accurate estimation of the control gain, with the concomitant suppression of chattering vibration and the prevention of redundant signal transmission. Literature [16] proposes the use of an enhanced adaptive super-twisting sliding mode observer, a technique that has been demonstrated to reduce the discontinuity of the control system, reduce the inherent chattering vibration, and effectively suppress the torque fluctuation. However, the method does not take into account the effect of disturbance on the system control performance.

The permanent magnet synchronous motor control system is affected by a variety of disturbance factors, including harmonic disturbance caused by unstable power supply voltage, torque fluctuation generated by sudden change of load torque, parameter variations triggered by temperature change and magnetic saturation, as well as systematic disturbance caused by delays in signal acquisition and measurement error, etc. The combined effect of these disturbances on the dynamic and steady-state performance of the system is a primary factor in the necessity to enhance the system’s performance in the disturbance rejection capacity. These disturbances jointly affect the dynamic and steady state performance of the control system, rendering the disturbance rejection problem a key challenge to improving the system’s performance. Therefore, this paper proposes to utilize a generalized proportional-integral observer (GPIO), which is developed based on linear observer theory. In comparison with the conventional frequency-domain disturbance observer, the generalized proportional-integral observer (GPIO) has the capacity to be applied to a range of disturbance types and can estimate the system’s internal states, external disturbances and their derivatives of each order in real time. In comparison with the traditional time-domain disturbance observer, the structure of the GPIO is relatively simple and can balance the dynamic response of the system and its noise suppression performance, making it more applicable in practice [17,18].

The paper proposes an enhanced adaptive controller, termed CCEASTA (enhanced adaptive super-twisting current-constrained algorithm), which integrates current constraint with an improved super-twisting sliding-mode algorithm within a non-cascade structure, building upon the foundation of reference [19]. The paper proposes a fourth-order generalized proportional-integral observer (GPIO) to accurately estimate the total collective disturbance and facilitate feed-forward compensation, thus improving the disturbance rejection performance of the permanent magnet synchronous motor (PMSM) speed control system. This composite controller, which combines the CCEASTA and GPIO, effectively limits the q-axis current of the PMSM to a manageable range, thereby safeguarding the hardware from overcurrent while maintaining the system’s dynamic performance [20]. Furthermore, the adaptive super-twisting sliding mode control employs a continuously differentiable switching function and speed-adaptive gain, thereby significantly reducing chattering and enhancing adaptability across the full-speed domain. The construction of the GPIO facilitates the estimation of disturbances and their higher-order derivatives, thereby enhancing both dynamic and steady-state performance, as well as disturbance resistance. This paper also incorporates practical non-ideal factors into the simulations, including sensor noise, discretization delay, inverter nonlinearity, and voltage saturation, to assess the feasibility and robustness of the proposed strategy within an engineering implementation context. This approach offers a safe and practical solution for high-performance permanent magnet synchronous motor drives.

2  PMSM Mathematical Model

It can be posited that, under the condition that the magnetic field is sinusoidally distributed, and with the assumption that hysteresis and eddy current losses are negligible, and further that the magnetic circuit is not saturated [21], the mathematical model of the surface-mounted permanent-magnet synchronous motor in the dq-rotating coordinate system can be expressed as follows:

(id∙i∙qωm∙)=(−RsLsnpωm0−npωm−RsLs−npψfLs0npψfJ−BJ)(idiqωm)+(udLsuqLs−TLJ)(1)

where id and iq are the stator currents in the d-axis and q-axis, respectively, and ud and uq are the stator voltages in the d-axis and q-axis, respectively, noting that np is the number of pole pairs, ωm is the mechanical angular velocity, Rs is the stator resistance, Ls is the inductance, TL is the load torque, ψf is the rotor magnetic chain, B is the coefficient of viscous friction, and J is the inertia of rotation.

The system equation of motion can be expressed as:

Jdωmdt=Te−TL−Bωm(2)

where Te is the electromagnetic torque.

The system electromagnetic torque equation can be expressed as:

Te=32npψfiq=K,iq(3)

where Kt is set to be the torque constant.

When establishing the mathematical model of the permanent magnet synchronous motor speed control system, in order to more accurately reflect the actual operating conditions, it is necessary to take into account the internal parameters of the control system as well as the time-varying influence of the external load torque. Therefore, its rotor motion equation should be constructed and described on this basis as:

dωmdt=−(BJ+Δa)ωm−(TLJ+Δc)+(npψfJ−Δb)iq0=aωm+bu−d(4)

In the system modeling, a=−BJ,b=npψfJ,u=iq represents the designed control input law, Δa, Δb, Δc are used to characterize the parameter variations generated by the motor during actual operation, and d integrates the aggregate disturbance [22] triggered by the above parameter changes in conjunction with external load torque fluctuations. thus, the following relation can be obtained:

d=Δaωm+Δbu+Δc+TzJ(5)

Current constraints are introduced to limit the q-axis stator current to |iq|.

Assumption 1: The system load torque variation satisfies the following conditions:

2TL+Bωm∗npψf<corTL<npψf(c−iq∗)(6)

where iq∗=Tz+Bωm∗npψf is set as the current balance point.

Remark 2.1: If the load meets the conditions of Assumption 1 in the unknown case, the current constraint still has a margin to suppress the load change, so c − i*q > 0 must be valid. c value is generally determined according to the rated current of the PMSM, which is usually two to three times of the rated current, and in this paper, c = 15A.

3  Enhanced Adaptive Super-Twisting Sliding Mode Current Constrained Controller Design

3.1 Current Constraint Function Design

In this paper, the barrier function is designed so that the constraint boundaries can be adjusted according to the state of the system while guaranteeing |iq|<c:

Φ(iq)={c22ln⁡(c2c2−iq2)+12iq2,|iq|<c∞,|iq|≥c(7)

Deriving the above equation yields the penalty term

ϕ(iq)=∂Φ∂iq=2c2−iq2c2−iq2iq(8)

Property 3.1: When condition |iq|→c− is satisfied, result ϕ(iq)→±∞ can be achieved and the current constraint function will form an infinity value.

Property 3.2: Condition Φ(iq)≥12iq2 satisfies the positive characterization.

Property 3.3: ϕ(iq) is continuously differentiable in the interval (−c,c).

To avoid singularities in the numerical calculations, a small constant ϵ>0 is introduced in the penalty term:

Φ~(iq)=c2+ε2ln⁡(c2+ϵc2+ϵ−iq2)+12iq2(9)

Remark 3.1: As the q axis current iq approaches the constraint boundary c both the barrier function Φ(iq) and its derivative ϕ(iq) exhibit a rapid increase in numerical stiffness and saturation of the control variable. To mitigate this issue, a small constant ε is introduced into the penalty term (refer to Eq. (9)) to restrict the maximum amplitude ϕ(iq) and prevent calculation overflow. In the actual simulation, ε is set to 0.001. At this value, ϕ(iq) at iq = 14.999 A is approximately 103, and the corresponding control increment remains within the voltage limit range, with a maximum voltage increment of about 5 V. If ε is excessively small (for instance, ε < 1 × 10−5), ϕ(iq) can exceed 105 near the boundary, potentially leading to high-frequency chattering of the control variable and numerical overflow. Conversely, if ε is too large (for example, ε > 0.1), the penalty effect near the boundary diminishes, resulting in a reduced constraint effect. The ε value chosen in this study strikes a balance between the strictness of constraints and numerical stability. Furthermore, the existing constraint term λ2ϕ(iq) is incorporated into the control law (15). As iq approaches c, this term becomes predominant in the control variable, compelling iq to decrease and thereby naturally preventing continuous control saturation. The simulation did not exhibit any numerical divergence or control saturation issues attributable to the barrier function.

3.2 Enhanced Adaptive Super-Twisting Sliding Mode Control Law Design

Define the integral sliding mode surface:

s=eω+λ1∫0teω(τ)dτ+λ2Φ(iq)(10)

where eω=ωm∗−ωm is the velocity tracking error and λ1,λ2>0 is a design parameter.

A continuously differentiable segmented switching function is used:

fα(x)={sign(x)(1−e−xα),|x|≥σxα(1−|x|2α+x26α2),|x|<σ(11)

where α>0,δ=α/2.

Property 3.4: fα(x) is second-order differentiable at x=0.

Property 3.5: |fα(x)|≤1,|fα′(x)|≤1/α.

Property 3.6: xfα(x)≥0, which guarantees positive characterization.

Design time-varying gain

{k1(t)=l1ωe(t)+γ1‖e(t)‖+μ1k2(t)=l2ωe2(t)+γ2‖e˙(t)‖+μ2(12)

where ωe=max(ωmin,|ωm|),e=[eω,iq]T,μ1,μ2>0 are lower bound constants.

Gaining adaptive laws:

{l˙1=η1(|s|−ε1l1)l˙1=η1(|s|−ε2l2)(13)

η1,η2>0 is the learning rate and ε1,ε2>0 is the forgetting factor.

3.3 Control Law Design

d-axis control law design:

Setting id=0, realized with PI control:

ud=kPd(0−id)+kId∫0t(0−id(τ))dτ+ωeLsiq(14)

q-axis control law design:

uq=uq,sq+uq,zω+uq,con=(Rziq+ωsLziδ+ωsψf+Lziqref)−Lz[k1|s|1/2fα(s)+∫0tk2fα(s)dτ]−λ2ϕ(iq)(15)

where ωe=npωm,i⋅qref=JKt(ω⋅m∗+λ1eω)+BKtωm+TLKt.

4  Generalized Proportional Integral Observer Design

4.1 Disturbance Modeling and Observer Structure

Modeling of aggregate disturbances:

Define the set total disturbance d(t) in conjunction with the control system:

d(t)=−KtJ(iqref−iq)−TLJ−BJωm+Δ(t)(16)

Let d(t) be expressed as a second order polynomial:

d(t)=a0+a1t+a2t2(17)

Remark 4.1: Eq. (17) approximates the lumped disturbance d(t) using a second-order polynomial model. This approach effectively captures slowly varying disturbances, such as gradual changes in load torque, ramp disturbances, including acceleration variations, and certain low-frequency harmonic disturbances, such as the sixth harmonic induced by the inverter dead zone, which are commonly observed in actual PMSM systems. Although this model may not accurately capture high-frequency random perturbations, such as measurement noise, or non-polynomial disturbances like step or square waves, the GPIO can still provide reliable estimations owing to its high-gain observer structure. The configuration of the observer poles effectively balances estimation accuracy and sensitivity to noise.

Extended state space model:

Defining the extended state vector x=[x1,x2,x3,x4]T=[ωm,d,d˙,d¨]T, the dynamic equations of the system can be expressed as:

{x˙1=KtJiq+x2x˙2=x3x˙3=x4x˙4=d(3)(t)(18)

Fourth-order GPIO design:

The design of the fourth-order generalized proportional-integral observer can be expressed as:

{z˙1=KtJiq+z2+β1(ωm−z1)z˙2=z3+β2(ωm−z1)z˙3=z4+β3(ωm−z1)z˙4=β4(ωm−z1)(19)

where z1 is the rotational speed estimate, z2 is the disturbance estimate, z3 is the first order derivative of the disturbance estimate, and z4 is the second order derivative of the disturbance estimate.

4.2 Parameter Integral and Stability Analysis

Generalized proportional-integral observer pole configuration design:

Set the observer error dynamic equation as:

e⋅=Ace+Bd(3)(t)(20)

where e=[e1,e2,e3,e4]T=[ωm−z1,d−z2,d˙−z3,d¨−z4]T, Ac=[−β1100−β2010−β3001−β4000].

The characteristic polynomial corresponding to the set error dynamic equation is:

det(sI−Ae)=s4+β1s3+β2s2+β3s+β4(21)

To obtain the desired dynamic response, the observer poles are configured as −ω0 (quadruple poles), i.e., the desired characteristic polynomial is:

(s+ω0)4=s4+4ω0s3+6ω02s2+4ω03s+ω04

Comparison of the coefficients gives: β1=4ω0,β2=6ω02,β3=4ω03,β4=ω04.

Optimal selection of observer bandwidth:

The observer bandwidth parameter ω0 selection requires a balance between dynamic response speed and noise rejection capability:

Construct the noise transfer function:

Gn(s)=z1σn=4ω0s3+6ω02s2+4ω03s+ω04(s+ω0)4(22)

Construct the disturbance suppression transfer function:

Gd(s)=z1δi=s3(s+ω0)4(23)

All things considered, ω0=30 rad/s was chosen to correspond:

β1=120,β2=5400,β3=108,000,β4=810,000

Proof of stability:

Theorem 4.1: For an observer error system, the estimation error is eventually bounded if d(3)(t) is bounded and ω0>0.

Proof: construct the Lyapunov function Ve=eTPe, where P satisfies AeTP+PAe=−Q. The derivation is obtained:

V⋅e=−eTQe+2eTPBd(3)(t)(24)

is given by |d(3)(t)|≤d¯3:

V⋅≤−λmin(Q)‖e‖2+2‖P‖‖B‖‖d¯3‖‖e‖(25)

When |e|>2|P||B|d¯3λmin(Q), V⋅e<0,the error is therefore bounded. The proof is complete. □

Feedforward compensation design:

The feed-forward compensated control volume is designed based on the disturbance estimates provided by the generalized proportional-integral observer:

uff=−JKtz2(26)

This feedforward term can be combined with the system control law to realize active suppression of disturbances and improve the overall robustness of the system.

5  Composite Control System Design

In order to simultaneously achieve accurate disturbance estimation and strict current limiting control, a composite control scheme is designed in this paper. The core of the scheme lies in the combination of enhanced adaptive super-twisting current constraint control (for direct current constraint) and generalized proportional-integral observer (for feed-forward compensation), and its overall control law consists of the following components:

uq=uq,eq+uq,sw+uq,con+uff=(Rsiq+ωeLsid+ωeψf+Lsiqref⋅)−Ls[k1|s|1/2fα(s)+∫0tk2fα(s)dτ]−λ2ϕ(iq)−JKtz2(27)

The overall block diagram of the control system designed in this paper is shown in Fig. 1.

Figure 1: Block diagram of the enhanced CCEASTA system with GPIO for PMSM.

6  System Stability Analysis

6.1 Main Symbols and Assumptions

To facilitate the subsequent analysis, we first describe the main notations used in the text (as shown in Table 1) and establish the necessary theoretical assumptions.

Main Symbols:

Hypothesis:

To complete the proof of stability of the system, we introduce the following three basic assumptions:

Assumption 6.1 (Disturbance boundedness): The system disturbance satisfies the following bounded conditions:

|ρ|≤Δ1,|δ|≤Δ2,|δ˙|≤Δ3(28)

where Δ1,Δ2,Δ3>0 is a known constant.

Assumption 6.2 (Nature of the current constraint function): The current constraint function ϕ(iq) is satisfied:

Positive characterization: ϕ(iq)iq>0,∀iq≠0.

Boundedness: 0<ϕmin≤ϕ(iq)≤ϕmax<∞,∀|iq|<c.

Continuity: ϕ(iq) is continuously micro-miniaturizable within (−c,c).

Boundary characteristics: lim|iq|→c−ϕ(iq)=±∞.

Assumption 6.3 (Switching function properties): Enhanced switching function fα(x) satisfied:

Continuity: fα(x) is continuous on R.

Boundedness: |fα(x)|≤1,∀x∈R.

Positive characterization: xfα(x)≥0,∀x∈R.

Boundary layer properties: when |x|<α, fα(x)≈xα can be obtained.

6.2 Derivation of Error Dynamic Equations

Define the sliding mode surface:

s=eω+λ1∫0teω(τ)dτ+λ2Φ(iq)(29)

Derivation of Eq. (29) yields the dynamic rate of change of the slip mold surface:

s⋅=eω⋅+λ1eω+λ2ϕ(iq)i⋅q(30)

Putting the q-axis voltage control law:

uq=(Rsiq+ωeLsid+ωeψf+Lsi˙qref)−Ls[k1|s|1/2fα(s)+∫0tk2fα(s)dτ]−λ2ϕ(iq)(31)

Substituting into the q-axis voltage equation:

Lsdiqdt=uq−Rsiq−ωeLsid−ωeψf(32)

Collation leads to the dynamic equation for the q-axis current:

diqdt=i′′−k1|s|1/2fα(s)−∫0′k2fα(s)dτ−λ2Lsϕ(iq)(33)

Dynamic equations for a sliding mode surface

Substitute Eq. (33) into Eq. (30) and organize the equation to obtain Eq. (34):

s⋅=ρ−λ2ϕ(iq)k1|s|1/2fα(s)+σ(34)

Among them:

ρ=e⋅ω+λ1eω+λ2ϕ(iq)i⋅qref−λ22Lsϕ2(iq)(35)

σ=−λ2ϕ(iq)∫0tk2fα(s)dτ(36)

Auxiliary variable definitions:

The derivation of σ yields:

σ⋅=−λ2dϕ(iq)dt∫0tk2fα(s)dτ−λ2ϕ(iq)k2fα(s)(37)

Let Eq. (38) be satisfied to obtain Eq. (39)

δ=−λ2dϕ(iq)dt∫0tk2fα(s)dτ(38)

σ⋅=−λ2ϕ(iq)k2fα(s)+δ(39)

6.3 Standard Super-Twisting Sliding Mode Control Law Transformation

Define state variables: x1=s,x2=σ.

Eqs. (34) and (39) can be represented as:

{x⋅1=ρ−λ2ϕ(iq)k1|x1|1/2fα(x1)+x2x⋅2=−λ2ϕ(iq)k2fα(x1)+δ(40)

Disturbance boundary analysis:

It follows from Assumption 6.1 and Assumption 6.2 that there exists a normal number Δ1,Δ2,Δ3 that makes:

|ρ|≤Δ1,|δ|≤Δ1,|δ˙|≤Δ3(41)

and ϕ(iq) is satisfied:

0<ϕmin≤ϕ(iq)≤ϕmax<∞(42)

6.4 Lyapunov Function Construction and Detailed Analysis

Define the vector:

z=[|x1|1/2sign(x1)x2](43)

where sign() is the sign function.

Lyapunov function selection:

A quadratic Lyapunov function is chosen:

V=zTPz=p11z12+2p12z1z2+p22z22(44)

where P is a symmetric positive definite matrix:

P=[p11p12p12p22]>0(45)

Parameterization of matrix P:

The super-twisting algorithm is set up according to its requirements for stability:

P=12[4k2+k12−k1−k12](46)

where k1,k2 is a positive constant. This matrix is satisfied:

1. Symmetry: P=PT.

2. Positive characterization: p11>0 and det(P)>0.

Calculating the determinant gives:

det(P)=14[(4k2+k12)⋅2−(−k1)2]=14(8k2+2k12−k12)=14(8k2+k12)>0(47)

6.5 Proof of Stability

Step 1: Calculate the time derivative of z.

By defining z1=|x1|1/2sign(x1), the derivation is obtained:

z⋅1=12|x1|1/2x1⋅⋅sign(x1)=12|x1|1/2x1⋅⋅x1|x1|=12|x1|1/2x1⋅(48)

where sign(x1)=x1/|x1|.

Substituting the x⋅1 expression yields:

z⋅1=12∣x1∣1/2[ρ−λ2ϕ(iq)k1∣x1∣1/2fα(x1)+x2](49)

By organizing |x1|1/2=|z1| and x2=z2, we can obtain:

z1⋅=−12λ2ϕ(iq)k1fα(x1)+12z2+ρ2∣z1∣(50)

Taking the derivative of z2 yields:

z⋅2=x⋅2=−λ2ϕ(iq)k2fα(x1)+δ(51)

Step 2: Matrix form representation

Write Eqs. (47) and (48) in matrix form:

z⋅=Az+B(52)

Among them:

A=[−12λ2ϕ(iq)k112−12λ2ϕ(iq)k20](53)

B=[ρ2|z1|δ](54)

Step 3: Calculate the Lyapunov function derivative

Derive for V=zTPz:

V⋅=z⋅TPz+zTPz⋅(55)

Substitute Eq. (52):

V⋅=(Az+B)TPz+zTP(Az+B)=zT(ATP+PA)z+BTPz+zTPB=zT(ATP+PA)z+2zTPB(56)

Step 4: Matrix ATP+PA calculation

Calculation ATP+PA:

ATP=[−12λ2ϕk1−λ2ϕk2120][p11p12p12p22]=[−12λ2ϕk1p11−λ2ϕk2p12−12λ2ϕk1p12−λ2ϕk2p2212p1112p12](57)

PA=[p11p12p12p22][−12λ2ϕk112−λ2ϕk20]=[−12λ2ϕk1p11−λ2ϕk1p1212p11−12λ2ϕk1p12−λ2ϕk2p2212p12](58)

The two equations are added together to obtain:

ATP+PA=[−λ2ϕk1p11−2λ2ϕk2p120012p12](59)

Step 5: Disturbance term analysis

The disturbance term 2zTPB can be expressed as:

2zTPB=2[z1z2][p11p12p12p22][ρ2|z1|δ](60)

Eq. (60) is calculated as Eq. (61):

2zTPB=(p11z1+p12z2)ρ|z1|+2(p12z1+p22z2)δ(61)

By the Cauchy-Schwartz inequality:

|(p11z1+p12z2)ρ|z1||≤(p11|z1|+|p12||z2|)Δ1|z1|=p11Δ1+|p12|Δ1|z2||z1|(62)

|2(p12z1+p22z2)δ|≤2(|p12||z1|+p22|z2|)Δ2(63)

Step 6: Finite time convergence conditions

To guarantee V⋅<0, the following conditions need to be satisfied:

Condition 6.1 (Main diagonal dominance condition):

−λ2ϕmink1p11−2λ2ϕmink2p12+12p12<0(64)

Condition 6.2 (Disturbance suppression condition): There exists ε>0 such that Eq. (65) holds.

−[λ2ϕmink1p11+2λ2ϕmink2p12−12p12]∥z∥2+p11Δ1+|p12|Δ1|z2||z1|+2(|p12||z1|+p22|z2|)Δ2≤−ε||z||2(65)

Step 7: Gain selection strategy

From the conditions Eqs. (57) and (58), the gain selection strategy is derived:

The lower bound of k1 is as in Eq. (66):

k1>|p12|2λ2ϕminp11+2k2p12p11(66)

The upper bound of k2 is as in Eq. (67):

k2>p124λ2ϕminp12=14λ2ϕmin(67)

The following conditions need to be met in order to suppress the disturbance:

k1>2Δ1λ2ϕmin,k2>Δ2λ2ϕmin(68)

The gain condition can be obtained as:

{k1>max(|p12|2λ2ϕminp11+2k2p12p11,2Δ1λ2ϕmin)k2>max(14λ2ϕmin,Δ2λ2ϕmin)(69)

Step 8: Finite time convergence proof

Theorem 6.1 (Finite time convergence): If the gains k1,k2 satisfy the condition Eq. (69) and there exists a constant γ>0,β∈(0,1), such that Eq. (70) holds:

V⋅≤−γVβ(70)

Then the state of the system converges to the slip mold surface T≤V1−β(0)γ(1−β) in finite time s=0.

Remark 6.1: Theorem 6.1 ensures that the system state converges to the sliding surface defined by s = 0 within a finite time frame. Based on the classical convergence analysis of the super-twisting algorithm [16], the convergence time T adheres to the following upper bound estimation condition:

T≤2V1/2(0)ε(71)

where V(0) represents the initial value of the Lyapunov function, and A is a constant associated with the gain. Based on the gain condition outlined in Eq. (62) and the derivative of the Lyapunov function presented in Eq. (64), we can derive

ε=2λmin(P)(k1−ρ+2)(72)

By appropriately increasing k1 and k2, it is possible to reduce the convergence time; however, one must consider the amplitude limit of the control input and the stability margin of the system. In practical parameter selection, ε can initially be estimated based on the desired convergence speed, after which the lower bound of the required gain can be derived from Eq. (62).

Proof: From Eqs. (51), (52) and (55), there exist positive constants α1,α2 such that Eq. (69) holds when condition (71) is satisfied.

V⋅≤−α1∥z∥2+α2∥z∥(73)

Since V=zTPz, there exists a constant κ1,κ2>0 satisfying Eq. (72)

κ1∥z∥2≤V≤κ2∥z∥2(74)

Substituting Eq. (73) into Eq. (72) yields:

V⋅≤−α1κ2V+α2κ1V1/2(75)

When condition V1/2>α2κ2α1κ1 is satisfied, one obtains:

V⋅≤−α12κ2V(76)

This indicates exponential convergence of V. When V is small, the dominant term is the V1/2 term and the system converges in finite time by the theory of finite time stability. The proof is complete. □

6.6 Current Constraint Function Validation Analysis

Construct the current constrained Lyapunov function:

Vc=12ln⁡(c2c2−iq2)(77)

Constructor properties are satisfied:

1. Positive characterization: Vc≥0, and Vc=0⇔iq=0.

2. Boundary characteristics: limtq|→c−Vc=+∞

Derive for Vc:

V⋅c=iqiq⋅c2−iq2(78)

Substituting into Eq. (33) for iq yields:

V⋅c=iqc2−iq2[iqref⋅−k1|s|1/2fα(s)−∫0tk2fα(s)dτ−λ2Lsϕ(iq)]=iqiqref⋅c2−iq2−iqk1|s|1/2fα(s)c2−iq2−iq∫0tfα(s)dτc2−iq2−λ2Ls⋅iqϕ(iq)c2−iq2(79)

Considering the last term of Eq. (77) yields Eq. (78):

iqϕ(iq)c2−iq2=iqc2−iq2⋅2c2−iq2c2−iq2iq=(2c2−iq2)iq2(c2−iq2)2(80)

When |iq|→c−: When condition |iq|→c− is satisfied, Eq. (80) is obtained:

lim|iq|→c−(2c2−iq2)iq2(c2−iq2)2=+∞(81)

Proof of Current Binding Guarantee:

Theorem 6.2 (Current constraint guarantee): If the initial condition ssatisfy |iq(0)|<c and λ2>0, the control system ensures that |iq(t)|<c holds for all t≥0.

Proof: Construct Eq. (77) as:

V⋅c=T1+T2+T3+T4(82)

Among them:

T1=iqiqref⋅c2−iq2

T2=−iqk1|s|1/2fα(s)c2−iq2

T3=−iq∫0tk2fα(s)dτc2−iq2

T4=−λ2Ls⋅(2c2−iq2)iq2(c2−iq2)2(83)

T4 dominates as |iq| approaches c. Let |iq|=c−ε, where ε>0 is a very small number, then Eq. (83) holds.

T4≈−λ2Ls⋅c2(2cε−ε2)2⋅(c−ε)2(84)

Therefore, there exists ε0>0 such that Eq. (76) holds when condition c−|iq|<ε0 is satisfied.

V⋅c<0(85)

By the principle of comparison Vc does not tend to infinity, so |iq(t)|<c holds for all t≥0. This certificate completes. □

6.7 Detailed Analysis of Adaptive Gain Stability

Gain Error Definition

Define the ideal adaptive gain l1∗,l2∗ and error:

l~1=l1−l1∗,l~2=l2−l2∗(86)

The adaptive law Eq. (13) is constructed as: