Fault compensation control of MIMO nonlinear systems subject to unknown control directions

This study is primarily focused on the issue of low-complexity prescribed performance fault compensation for multi-input and multi-output uncertain nonlinear systems with actuator failures, coupling states, and unknown control directions. First of all, proper logarithm-type error conversion functions and smooth orientation functions are linked to design the continuous control signals in the state feedback controller. Then, based on the idea of proof by contradiction, it is shown that the state errors converge to a predictable compact aggregate at the definite rate. Meanwhile, the boundedness of any closed-loop signals might be guaranteed. The numerical and actual simulation examples are delivered to demonstrate the effectiveness of the developed control strategy at last.


Introduction
With the rapid development of modern industrial technology, composite engineering systems with uncertain nonlinearities, strong state couplings, actuator failures, and external disturbances are more and more widely used. Because of these difficulties, numerous research findings are committed to the single-input and single-output (SISO) nonlinear systems like [1,2]. It is worth noting that lots of actual industrial systems contain multiple inputs and outputs, such as flexible multi-link manipulators [3,4], satellite [5,6], multiple under-actuated autonomous surface vehicles [7], and permanent magnet synchronous motor [8,9]. Hence, for the past few years, a large number of scholars have been working on the robust controller design of MIMO uncertain nonlinear systems. In [10], Chen, M. et al. developed a robust constrained control scheme for MIMO nonlinear systems based on backstepping control technology and sliding mode disturbance observer (SMDO). What's more, he also studied the robust adaptive neural network (NN) control problem of a class of uncertain MIMO nonlinear systems with input nonlinearities and unknown control coefficient matrix moments in [11]. In [12][13][14], regarding some systems with unknown control directions, the Nussbaum gain technique and control laws were merged into the controller model. Specifically, [14] considered the tracking control problem of a class of MIMO nonlinear systems with unknown directional control gain and input saturation. A command filter adaptive NNs method aimed at MIMO by designing virtual controllers and error compensation signals. In addition, the Nussbaum type function was used to process the unknown direction control gain. However, as Oliveira et al. pointed out in [15][16][17], the practical value of this method was controversial owing to the inherent lack of robustness, large fluctuation peaks, and poor transients. To avoid these matters, adaptive control mechanisms were applied to observe the unknown parameters by referring to [18][19][20][21][22]. Their shortcoming was that the real-time updating of online learning parameters greatly increased the computational complication of the control algorithm, which made the control machine difficult to be implemented in engineering. Furthermore, see [2,[23][24][25][26] and reference therein, the implementation of backstepping method needs to calculate the derivatives of virtual control rate, it is quite complicated and easy to cause relatively large errors.
On the one hand, in order to dodge the trouble of virtual control mechanisms, more approximation techniques were adopted. In [27], a neural network adaptive robust control (NNARC) method was proposed. It exploited the multi-layer neural network to approximate the unknown nonlinear term. Further, better model compensation and improved performance were obtained. With the help of fuzzy adaptive observer making up for unmeasured states, the authors of [28] transformed the tracking errors into new virtual error variables, and a fuzzy adaptive output-feedback control method was offered. On the other hand, the computational complexity has been reduced by adopting dynamic surface control (DSC) technology. In [29], a first-order filter was introduced which overcame the phenomenon of parameter expansion caused by traditional backstepping. Besides, the assumption of the fault boundary was eliminated. From this foundation, reference [30] considered the actuator failures of MIMO uncertain systems. In addition, DSC was applied to FTC, and then, a fuzzy fault-tolerant control (FFTC) mechanism was raised. In fact, there have been many research results on FTC related to nonlinear sys-tems [31][32][33][34]. In [34], the authors derived a distributed FTC scheme by combining neural network, backstepping, DSC, and algebraic graph theory. It was usually used to settle the puzzle of output tracking control based on pure feedback multi-agents. The intelligent approximate technique and DSC method could effectively handle with tracking control problem of uncertain nonlinear systems during the backstepping process. However, there still exists a relatively complex recursive design with adaptation mechanism or parameter updated laws. Also, the corresponding stability analysis results are only confined in the local compact sets.
Although there have been a lot of improvements, none of the above approaches are able to specify transient performance, only the convergence of tracking error to residual set is established. Exactly, the involvement of PPC in [35][36][37] compensated for this deficiency. Because it can not only cause the tracking errors converge to a predictable compact set, but also ensure that the convergence rate is not less than a predetermined value. Correspondingly, just as Theodorakopoulos analyzed in [38], the simplification in structure and calculation still made PPC more widely used. For example, [39] focused on solving PPC of SISO strictfeedback nonlinear systems. The error transformation functions were associated with the specific Nussbaumtype functions. Additionally, a trigonometric type error transformation function was come up in [40,41].
Inspired by the above discussions, this paper studies the FTC and prescribed performance control problems for the type of MIMO uncertain nonlinear systems with actuator failures and unknown control directions. For the research results given, the major contributions of this paper are summarized as below: (1) Here, we consider the more general uncertain MIMO nonlinear systems with actuator failures, it is worth mentioning that [41] and [42] are special cases of this paper. (2) Based on the idea of inverse hyperbolic tangent function, an appropriate logarithm-type of error conversion function cooperated with directional function is proposed to realize PPC for the MIMO system. This is different from the tan-type commonly used in [39][40][41]. (3) By the programming of the control form, neither the approximation strategies in [27,28] nor the higherorder derivatives of reference signals in [42][43][44] are used. As a result, the controller is simpler than com-parable results in structure and calculation. Besides, the adaptive parameters are not any more needed to be updated compared with [19,20], which both reduce the computational cost, and ensure that the state error converges to an arbitrarily small residual set at a definite rate not less than the predetermined value.
The remaining part of this paper is organized as follows: In Sect. 2, the problem statement and several related assumptions are introduced. Meanwhile, a design procedure of the controller is given. Section 3 presents the control process and stability analysis. The numerical simulation results are provided in Sect. 4. In the end, Sect. 5 draws the conclusions.

Problem formulation and preliminaries
Take the following class of MIMO uncertain nonlinear systems with actuator failures and unknown control directions into account: . . where n i are the state variables of the systems; u i, f ∈ R and y i ∈ R denote, respectively, the system control input and output; both f i, j (·) :∈ R i → R and g i, j (·) :∈ R i → R are the unknown nonlinear functions and locally Lipschitz in x i, j ; furthermore, the perturbation term d i, j (t) ∈ R is piecewise continuous in t.
The sign of g i, j (·) plays the role of control direction. The actual control input u i, f ∈ R describes as follow: where u i (t) is a control input to be designed, λ i (t) and δ i (t) are the time-varying partial loss of effectiveness and the float faults.

Remark 1
The unknown nonlinear functions f i, j (·) and g i, j (·) satisfy the local Lipschitz condition, so that the existence and uniqueness of the solution of the system (1) are guaranteed. At the same time, it also shows that the function is absolutely continuous on the closed interval. This leads to the boundedness theorem of continuous functions, which is applied in the subsequent stability analysis.
Remark 2 For the systems with unknown control direction in [47] and [48], it is assumed that . . , m and g i,n i (x) = g n i with g n i = 0 being a constant, whereas all the control direction functions g i, j (·), j = 1, 2, . . . , n i are allowed to be unknown in this paper. In addiction, more universalities on given system comparing with [40] and [42]: (1) just constrain |g i, j (·)| to have a lower bound; (2) the unknown nonlinear function f i, j (·) can be any continuous function, it does not require boundaries.

Remark 3
As mentioned in Assumption 3, the target trajectory and its first-order derivative are bounded that mean fewer restrictions than [42] and [43]. Because the controller design strategy combines error transformation and smooth orientation functions, it is no longer needed to obtain higher derivatives of r i (t), i = 1, 2, . . . , m.
The ultimate control objectives are to design the continuous virtual control signals and control law to achieve: (1) the closed-loop system (1) is globally stable in this sense that each reference signal is uniformly ultimately bounded; (2) state errors are limited within prescribe performance functions; (3) control outputs of the system can track the given reference signals r i (t), i = 1, 2, . . . , m as close as possible.

Controller design
In this subsection, the state feedback controller is contrived to make the above-mentioned control objectives come true. First of all, define the state errors as where r i (t) is the user-defined tracking trajectory and α i, j expresses the continuous virtual control signal which will be defined later. The prescribed performance functions q i, j (t) were selected to restrict state errors, so that where set up 0 < q For ∀t ≥ 0, define error conversion function as where e i, j and q i, j meet the above definitions (4) and (5).

Remark 4
The design inspiration of the error conversion function (6) comes from the reverse hyperbolic tangent function tanh −1 (x) = 1 2 ln 1+x 1−x by referring to [45] and [46]. Owing to the logarithmic function limits the domain, there will be |x| < 1 − ε, 0 < ε < 1 hold. For the same reason, there exists a constant Next, combine the error conversion function ξ i, j with the smooth orientation function N i, j (·), the continuous virtual control messages and control law are described as where γ i, j represents the positive control gain. Meanwhile, the function with a i, j and b i, j being two positive constants. For example, Remark 5 The introduction of the virtual control rate α i, j is to ensure that the tracking error converges to a preset bound and that all closed-loop signals are bounded. Compared to [19], we did not introduce an adaptive mechanism, so its structure will be relatively simple. Moreover, it is mainly affected by four param- i, j , and μ i, j , which can be adjusted according to the tracking effect and the degree of error limitation in the simulation.

Remark 6
As we all know, for the logarithmic functions, the function value approaches minus infinity as the independent variable verges to the original point; the function value approaches positive infinity as the independent variable verges to the positive infinity. Therefore, from Eqs. (6)- (8), thinking about the state error e i, j could be positive or negative, an important conclusion is drawn as where i = 1, 2, . . . , m, j = 1, 2, . . . , n i .

Stability analysis
In this section, a sufficient cause for the derivative bounded of the continuous virtual control signal is given by proving Lemma 1. Furthermore, employ the usual proof by contradiction in mathematics: (1) demonstrate that all the closed-loop signals maintain bounded; (2) the control output can track the target trajectory by means of the proof process of Theorem 1. Proof From equation (7), the derivative of the continuous virtual control signal can be obtaineḋ Differentiating (6) leads tȯ By hypothesis ξ i, j = ln To sum up, q i, j ,q i, j and 1/q i, j are bounded, thenξ i, j ∈ L ∞ . Every component ofα i, j is bounded; Lemma 1 holds. A complete proof of Lemma 1.
To limit the state errors to the prescribed performance bounds in the form of By looking for the opposite condition, there exists at least one state error variable e (k) i, j (t l ), it follows that where t l < t l+1 . Suppose that t 1 represents the first moment that (12) is violated. Therefore, when t ∈ [0, t 1 ), one obtains It means that e i,1 , e i,2 , . . . , e i,n i , i = 1, 2, . . . , m are bounded over [0, t 1 ). In view of this, a state error variable e (k) i, j (t) satisfying where t − 1 is the left limit of t 1 . As the state error e i, j close to its prescribed performance limits q i, j in two directions, it naturally holds that In the meantime, the derivative of q i, j can be found by (5). Rewrite (16) as follows where i, j with i = 1, 2, . . . , m and j = 1, 2, . . . , n i . (1) with actuator failures and unknown control directions, compatible with Assumptions 1 and 2. Given initial condition |e i, j (0)| < q i, j (0), and the target track obeys aforementioned Assumption 3, the advanced state-feedback control strategy (7) fulfills the following conditions:

Theorem 1 Discuss one type of MIMO uncertain nonlinear systems
(1) the system control output could track the reference signal with the prescribed performance; (2) the whole signals of the closed-loop system remain bounded for t ≥ 0.
One can easily get the variable φ i,2 is bounded in the interval [0, t 1 ). Further, combine with (18) and (19), there are conclusions that The contradiction between (17) and (22) implies Step j (3 ≤ j ≤ n i − 1): The same procedure as Step 2, φ i, j ∈ L ∞ can be ensured. At the same time, there exists a positive constant c i, j < q It is clear that the error transformation function ξ i, j is bounded.
Because it contradicts (17) when j = n i , one has i,n i being a constant. This indicates the function ξ i,n i , i = 1, 2, . . . , m is  bounded over [0, t 1 ). Combining (21), (23), (24), and (26), it follows that which contradicts (15). So the hypothesis (13) is not true, as a result (12) is true. Now, let's testify the closed-loop signals are bounded when t ∈ [0, ∞). Redo Step 1 − n i by substituting (12) into (14), the function φ i, j maintain bounded. Also, it is given by Similarly, state error |e i, j | cannot get to its prescribed performance boundary q i, j for any t ≥ 0, or in other words, there exists a constant 0 < c i, j < q (∞) i, j , such that |e i, j | ≤ q i, j − c i, j , i = 1, 2, . . . , m, j = 1, 2, . . . , n i . (29) In summary, the whole signals of the closed-loop system are bounded. This completes the proof.

Remark 7
The reason that proof by contradiction sets up because when e i, j − q i, j reaches the origin from left, the limit ofė i, j is greater than a constant in (17). Thus, it is impossible that the lower limit ofė i, j equal to minus infinity in (20). The same reason for the e i, j + q i, j when it gets to the origin from right. Therefore, (28) conflicts with (17), the assumption (12) is invalid, i. e., the state errors can be limited in the prescribed performance functions.

Remark 8
It shows that all the closed-loop signals keep bounded by proving Theorem 1. For clarity, the specific processes are summarized as follows: (1) in the case of i = 1, due to (29) holds, so the tracking error e i,1 is bounded, and then, control output y i is bounded by (4); (2) because the continuous virtual control signals and control rate defined by (7) are bounded, so the control input is bounded; (3) due to (4) and (29), the states of the system are bounded.

Simulation studies
This section gives complex numerical examples and practical examples of permanent magnet synchronous motors (PMSM). Through the simulation results, whether in theory or in practical applications, the effects and preset errors of the controller can be reflected.

Numerical example
Here, we consider a third-order uncertain MIMO nonlinear system with actuator failure and unknown control direction: anḋ where x = [x 1,1 , x 1,2 , x 2,1 , x 2,2 , x 3,1 , x 3,2 ] T , u i, f and d i, j denote the system state variables, control inputs and external disturbances. Next, choose the unknown smooth nonlinear functions f 1, The number of pole pairs P 3 The moment of inertia J 0.08kg · m 2 The coefficient of friction B 0.02N · m/(rad/s) In addition, the tracking signals r i (t) are taken as r 1 (t) = 0.2 cos(t), r 2 (t) = 0.2 cos(t) + 0.2 sin(t) and r 3 (t) = 0.4 sin(t). By the prescribed performance bounds of the state errors (5), the simulation parameters are selected as:  (7), the control gains are given as γ 1,1 = 10, γ 2,1 = 12, γ 3,1 = 15, γ 1,2 = 8, γ 2,2 = 20, γ 3,2 = 20, respectively. Figures 1, 2, 3, 4, 5 and 6 are the simulation end results of above MIMO system (30)- (32), which realized by the state feedback controller designed in Section 2. Although the system has uncertainties and disturbances, it can be obtained by Theorem 1: (1) From Figs. 1, 2 and 3, the state errors e i, j , j = 1, 2 of subsystem i, i = 1, 2, 3 are strictly limited to the prescribed performance bounds ±q i, j ; (2) The output signals y i can track the reference signals r i , i = 1, 2, 3 very well in Fig. 4; at the same time, due to the better tracking effect, it is ensured that the state error e i,1 = y i − r i can converge quickly;  Fig. 6).
From the above, it is clear that the prescribed transient and steady-state tracking capability of the uncertain MIMO unknown nonlinear system could be guaranteed. In addition, all closed-loop signals are bounded.

Remark 9
This paper uses a relatively low-complexity method to control the state errors of the system. It can only guarantee that the errors will eventually converge to a small area, i. e. |e i, j | ≤ q i, j −c i, j , i = 1, 2, . . . , m, j = 1, 2, . . . , n i . And this interval can be infinitely small. However, complete asymptotic tracking has not been achieved. On the contrary, some other methods (for example: backstepping method) seem to be able to achieve the effect of asymptotic tracking. But its entire design process will be relatively complicated, especially in the derivation of virtual functions, it is easy to produce exponential explosion. Therefore, we will actively explore methods that can achieve both low-complexity PPC and asymptotic tracking of state errors. This is also a very serious difficulty facing the research field.

Application example
Next, in order to more directly prove the practical value of the research method proposed in the paper, a practical application simulation example [39] was used: where the state variable x = i d , i q , ω r T and the control input u = u d , u q T . Concretely, i j represents the armature current of the j-axis component, ω r is the rotor speed, and u j stands for the armature voltage of the jaxis component ( j = d, q). The remaining parts f (x), , and c are expressed as follows: In the specific simulation, according to the known variable information and the relationship between them, the above differential equations are disassembled to become a MIMO system conforming to system (1), as shown below: anḋ where x 1,1 = i d , x 2,1 = ω r , x 2,2 = i q , u 1 = u d , u 2 = u q . The descriptions and values of the remaining motor parameters are shown in Table 1.
Finally, based on the given initial value x(0) = [−0.2, 0.1, 0.15] T , the simulation results are shown in Figs. 7, 8, 9, 10, 11, 12 and 13. Although the actual system is complex and has disturbances, it can be seen that the proposed control method not only guarantees output tracking with preset performance (see Figs. 7,8,9,10), but also enables all the closed-loop signals to be bounded in the control process (see Figs. 11,12,13).

Conclusion
In this paper, a lower complexity prescribed performance and FTC approach for MIMO uncertain nonlin-ear systems accompanied by coupling states, actuator failures, and unknown control directions are proposed. Utilizing state feedback control, the intermediate control signals which combining error transform functions and smooth orientation functions are designed to make the state deviations converge to a predictable compact aggregate and guarantee the boundedness of possessive closed-loop signals. Ultimately, two simulation examples turn out to show the effectiveness of the proposed control scheme. Next, we will consider that the system will reach a stable state as soon as possible within a limited time and try to achieve the result of asymptotic tracking. Data availability Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.