Robust Temporal Logic Motion Control via Disturbance Observers

The existing motion control systems are largely concerning given reference tracking or stabilization. High-level of autonomy within robotics and autonomous systems demands new motion control methods to realize more complex goals rather than given references, while maintain involved safety conditions. This article tackles the robust temporal logic motion control problem for a class of disturbed systems. A disturbance observer (DOB) is used for unmatched disturbance estimation, and signal temporal logic (STL) formulas are introduced to express complex sequential tasks. For atomic temporal logic formulas, in order to maximize its robust semantics, a continuous feedback composite controller is constructed by utilizing the transient characteristics of the prescribed performance function and the DOB. We then present a suitable switched control strategy to guarantee the satisfaction of STL specifications consisting of conjunctions of the atomic temporal logic formulas. When the given STL formula is satisfied, the system robustness specified by the user meets the temporal logic specifications. Experimental results are illustrated to verify the effectiveness of the proposed method.

For LTL and MITL, existing control methods mainly rely on a finite abstraction of model dynamics and a language equivalent automata [5] or timed automata [6]. The controller is developed by solving a game over the product automata [7]. Recently, STL is developed as a new temporal logic language to describe continuous goals or specifications [3], [4]. The fundamental logical predicates of STL depend on the real-valued functions containing system states and the STL formulas include strict time limits. Due to the advantages of treating real valued signals explicitly [3] and allowing the use of qualitative semantics [8], the control design under STL specifications has attracted extensive attentions [9], [10], [11], [12], [13], [14], [15], [16], [17]. Due to the distinct feature of real-valued signal, the automota-based control methods for LTL and MITL are not available for systems subject to STL [9]. Due to the existence of noncausal, nonsmooth semantics and nonconvex, the formal methods-based motion control of systems subject to STL specification is quite challenging [10]. Existing control approaches for systems under STL specifications mainly include optimization-based approaches [12], [13], control barrier function approaches [9], [14], [15], reinforcement learningbased control strategies [16]. Optimization-based approaches are mainly applicable to discrete-time systems, and the core idea is to encode STL formulas as mixed-integer constraints, and then, the satisfying controller is obtained by solving a series of optimization problems. Control barrier function method can be used for continuous-time systems, the idea is to design a control barrier function that encompasses STL specifications, and then, derives the controller by solving a quadratic programming problem [9], [14], [15]. Although this method is more computationally efficient, how to design the corresponding control barrier function is still an open question. Besides the optimization-based control approaches, recent works (e.g., [10], [17], [18], and [19]) established a new research direction by embedding the prescribed performance control (PPC) into the hybrid system framework to meet the STL tasks, which can be directly used for continuous-time systems with benefits such as computational efficiency and considerable robustness. The controller developed in the aforementioned literatures has the advantage of simple structure, while the uncertainties in the system are suppressed by using prespecified boundaries, which are usually not able to provide sufficient control accuracy for systems suffering from significant disturbances or uncertainties.
Although various control methods have been developed for the dynamic systems subject to STL specifications, guaranteeing robustness under disturbances and uncertainties is practical and important, but quite challenging. For most motion control systems, multifarious uncertainties, such as load disturbances, sensor noise, nonlinear unknown friction, always bring undesirable influence on the performance specifications [20], [21]. As a direct and effective method, disturbance observer (DOB)-based control (DOBC) can effectively compensate the influences of model uncertainties and unknown disturbances in motion control systems [22], [23], [24], [25], [26]. The main advantages of DOBC are that the robustness of the controlled system can be guaranteed without destroying the nominal system performance, and the disturbances can be completely compensated via DOBC if they can be accurately estimated via an adequate DOB [27]. Despite the aforementioned advantages, most existing results are only available for matched disturbance compensation. However, unmatched disturbances widely exist in practical systems, for example, see the PMSM system [28], [29], [30], the MAGLEV suspension vehicle [31], to name but a few. Since compensating for unmatched disturbances is of great significance in both and engineering applications and theoretical, some literature have solved such problems by using DOBC, for instance, see [21], [27], [29], [30], [32], and [31].
This article simultaneously tackles the STL specifications and matched/unmatched disturbance rejection problems for motion control systems. Different from the recent works [10], [17], [18], [19], this article develop a robust temporal logic motion control (RTLMC) approach by virtue of the backstepping-based PPC method. PPC [33], [34] is a funnel-based feedback control method that considers both the transient and steady-state behavior of the tracking error. A user-defined performance function prescribes a desired behavior to this error that is subsequently achieved by a continuous feedback control law. For the STL specifications, we transform the problem into a PPC problem, and use an appropriate method to replace the tracking error with the corresponding robust semantics. The desired temporal behavior of the error curve can be implemented by a user-defined prescribed performance function (PPF) when the transformed system is stabilized via adequate control design. Specifically, for atomic temporal logic formulas, we utilize the PPF combined with DOBC to construct a composite controller, which can maximize the robust semantics. Subsequently, the RTLMC is developed to meet STL formulas consisting of such atomic temporal logic formulas. Different from [10], [17], [18], and [19] where only the intermediate control signal is selected as the form in [10] and [34], the disturbance observation technique is introduced in this article that greatly improves the control accuracy under disturbances. As compared with many conventional backstepping design methods, the estimates of disturbances are embedded into the virtual control signal to compensate the influences caused by external disturbances.
To conclude, the main contribution and novelty of this article is to develop a novel control strategy for a class of motion control systems with second-order dynamics, which simultaneously fulfilling the STL specifications and compensating undesirable effects of matched/unmatched disturbances. The rest of this article is organized as follows. Section II introduces preliminaries and problem formulation. Section III presents the main results of this article. Section IV gives the experimental results to validate the effectiveness of the proposed RTLMC method. Finally, Section V concludes this article.
Notations: R + , R ≥0 , and R denote the positive number set, nonnegative real number set, and the real number set. R n is the n-dimensional vector space over the real number set R. and ⊥ represent logical true and false, respectively. The arguments of functions are sometimes simplified, for example, a function f (x(t)) is denoted by f (x), f (·) or f .

A. Preliminaries
To begin with the control design, we first present the following preliminaries regarding STL and PPC.

1) Signal Temporal Logic (STL) [3]
: STL determines whether a predicate μ is true or false by calculating a predicate function h μ (x) : In this article, we focus on the following basic STL formulas: where ψ 1 and ψ 2 are STL formulas, and ∧ and ¬ are the Boolean conjunction and negation operators. G [a,b] and F [a,b] are always and eventually operators over the interval [a, b], respectively, with a, b ∈ R ≥0 . ψ, ψ 1 , and ψ 2 are called nontemporal formulas. θ and φ are called temporal formulas due to the existence of always and eventually operators. In addition, formulas of (2b) are called the atomic temporal formulas and (2c) are referred to sequential formulas. Let x(t) : R ≥0 → R be a continuous time signal, (x, t) |= φ denotes that the STL formula φ are satisfied by the part of the signal starting from t. The definitions of STL semantics [3] are as follows: Example 1: We consider an omnidirectional robot system as an example to explain the significance of these STL formulas. Let T indicates the position and x 3 (t) denotes the orientation. The robot is implementing a task to reach the target position [5,5] T within ten time steps with allowable 1 unit deviation, and keeps the orientation of 45°w ith allowable 8°deviation. This complex task is difficult to represent in existing control systems, but can be expressed

2) Prescribed Performance Control (PPC) [33], [34], [35]:
PPC is a funnel-based feedback control strategy that considers both the steady and transient state behavior of the tracking error. A user-defined PPF prescribes a desired behavior to the error that is subsequently fulfilled by a continuous feedback controller. For instance, in order to guarantee the steady-state and transient performance of the error E(t), a positive monotonic decreasing smooth function ϕ(t) : R ≥0 → R + satisfying lim t→∞ ϕ(t) = ϕ ∞ and ϕ(0) = ϕ 0 will be chosen as the PPF. Without losing generality, we chose the PPF as ϕ(t) = (ϕ 0 − ϕ ∞ )e −lt + ϕ ∞ with l, ϕ 0 , and ϕ ∞ are positive valued parameters to design.
The objective of PPC is to design a continuous feedback controller such that E(t) strictly satisfies where δ ≥ 0 is positive constant to assign. The parameters ϕ 0 and δ should be selected such that −ϕ 0 < E(0) < δϕ 0 in the context of PPC. To solve the control problem with restriction given by (3), an error transform can be introduced to formulate the constrained control problem of (3) into an equivalent unconstrained one. In order to achieve the aforementioned purpose, a transform function ξ(ε(t)) of the transformed error ε(t) is defined, which has the following properties: is monotonically increasing, the transformed error ε(t) can be expressed in the following form: If ε(t) can be stabilized to be bounded, then (3) is guaranteed. In summary, the control problem for E(t) with the condition (3) is equal to stabilize ε(t) in (4).

B. Problem Formulation
We consider a class of the second-order motion control system, which is depicted by where η = [η 1 , η 2 ] T ∈ R 2 , y ∈ R, and u ∈ R are the system state, system output, and control input, respectively. f i ∈ R, g i ∈ R, and d i ∈ R, i = 1, 2 are known nonlinear continuous functions, positive constants, and external disturbances, respectively.
To begin with, the disturbances are supposed to satisfy the following assumption.
Assumption 1: The disturbances in (5) are bounded, and the derivative of d i has a boundary d * i , which satisfies Remark 1: In practice, many motion control systems can be expressed as system (5), such as 1-DOF mechanical system [36], rotary manipulator system [37], and permanent-magnet linear synchronous motor system [38].
For the motion control system (5), our goal is to design an RTLMC law such that (y, 0) |= θ, where θ is a sequential formula specified in (2c).

III. ROBUST TEMPORAL LOGIC MOTION CONTROL (RTLMC)
This section presents the design procedure of the proposed RTLMC law for the motion control system (5) and provides rigorous stability analysis.

A. Design of DOBs
To obtain the estimations of external disturbances in the system (5), we introduce the following nonlinear DOB [23]: where D i is the upper bound of e i (t).

2) Encode of an Atomic STL:
To facilitate the RTLMC design, an encoding mechanism is proposed for an atomic temporal formula φ. Let ψ be the nontemporal formula appearing in temporal formula φ, r ≥ 0 is a robustness measure, ρ max > r is a design parameter, then we achieve by specifying an expected temporal behavior to ρ ψ (y) through the prescribed boundary and designing the performance function ϕ(t).
On the basis of Assumption 2, if (10) holds, the atomic STL specification (9) is satisfied with the following well-designed parameters [17]: Remark 2: The selection of parameters (11)-(15) provides a theoretical guarantee for r < ρ ψ (y, 0) < ρ max ∀t ≥ t * . Theoretically speaking, t * , ρ max , r, and ϕ(t) can be selected freely. However, in practical application, selecting small t * may result in extremely large control input, which may be strict due to the platform restrictions.

1) Design of RTLMC:
In this subsection, an RTLMC law is developed for the motion control system (5). By virtue of the design principal of PPC, we define E(t) = ρ ψ (y) − ρ max , then (10) is equivalent to which deduces (3) by letting δ = 0. The transformed error can be defined as If ε(t) can be stabilized to be bounded, then (16) is guaranteed, thus (10) is satisfied.

Step 2 -Development of RTLMC law: Let
Taking the time derivative of V 2 along (5) yieldṡ The RTLMC law with an atomic predicate u is designed as ∂d 1 λ 1 ) 2 with a constant l 2 > 0. By using Young's inequality and Assumption 1, substituting (22) into (21), one haṡ

2) Proof of Performance Guarantee:
The proof of rigorous performance guarantee of the proposed RTLMC strategy is provided in this subsection, which is summarized by the following theorem.
Proof: See Appendix. Remark 3: Although the RTLMC law is derived by complex mathematical methods, it should be highlighted that most of the analytical process, especially the proof part of Theorem 1, is for the rigorous of the theoretical results and does not get involved in the control implementation. It can be concluded from (22) that the proposed RTLMC law is mainly composed of a composite controller and two DOBs. The DOBs are designed to estimate external disturbances, and then, the estimated values are employed to compensate for the influence of disturbances. Its simple and clear structure is favorable for control implementation. It should be highlighted that the proposed RTLMC algorithm can be extended to high-order nonlinear system (e.g., high-order motion control system [21], MIMO system [35], Euler-Lagrange system [39], etc.) by integrating PPC for high-order nonlinear systems [33], [34], [35] into the proposed RTLMC.
Remark 4: The unmatched disturbances are handled via a composite backstepping design approach. First, to obtain the estimations of external disturbances in system, we introduce the nonlinear disturbance observe (6). Next, the estimates of disturbances by (6) are introduced into the virtual control law to compensate the effects caused by unmatched disturbances. In Step 1 of the design of RTLMC, the virtual control law effectively compensates the effects caused by unmatched disturbance d 1 . Since disturbance d 2 and control input are in the same channel, the effects caused by d 2 can be directly compensated by the controller.

A. Mathematical Model of the PMSM
We use a noncascade structure instead of the traditional cascade of current loop and speed loop. Usually, the d-axis reference current i * d is set as i * d = 0 to mitigate the couplings between currents and angular velocity. Here, the d-axis current-loop adopts a PI controller. The dynamic model of a surface-mounted PMSM under consideration is depicted by [29], [32] where L is the stator inductance, i q and u q are the stator currents and stator voltages of q-axes respectively. R s is the stator resistance, ω is the angular velocity, T L is the load torque, ψ f is the rotor flux linkage, J is the moment of inertia, B is the viscous friction coefficient, and n p is the number of pole pairs.

B. Experiment Verification
This subsection established the experiment setup system as shown in Fig. 3 to validate the presented RTLMC method. The platform consists of a host PC, an integrated controller board, and two identical PMSMs. One of the two PMSMs is the controlled object and the other is to apply load torque. By using MATLAB/Simulink, different types of load torques can be generated. The integrated controller board includes an   ETL-PCI-662 DSP motion control board and dual processor. The system parameters are given as R = 0.72Ω, L = 0.4 mH, B = 3.5 × 10 −4 N · m · s/rad, n p = 4, ψ f = 8.1 × 10 −3 Wb, and J = 2.824 × 10 −3 kg · m 2 . In order to verify the effectiveness of the presented RTLMC method, an improved funnelbased feedback controller (FBFC) based on [18] and [34] is adopted as a benchmark PPC, in which the uncertainties are suppressed by using prespecified boundaries.

1) Control Performance Under Parameter Uncertainty:
In this case, we consider that the inertia J undergoes significant changes, that is, J = 7.06 × 10 −4 kg · m 2 is the nominal value, the control performance of the two controller are tested in the absence of external load torque.
The system responses of ω, i q , and robustness curves ρ φ are shown in Fig. 4(a)-(d), respectively. As shown in Fig. 4(a) and (d), within the allowable error range, under the action of both controllers, the speed can reach the specified speed range within the specified time and the ideal segmented speed regulation effect is achieved. However, the FBFC approach delivers poor speed regulation performance, which is shown by considerable tracking errors, while the proposed RTLMC approach obtains far better speed regulation performance. It can be seen from Fig. 4(b) that the proposed RTLMC law is well tuned to achieve a satisfactory nominal control performance.

2) Robustness Performance Against Various Load
Torque Disturbances: Next, we analyze the performance of the presented RTLMC method under the following various load torque disturbances. 1) Step torque: T L = 0.3 N · m during 5 ≤ t ≤ 8 s.
2) Sinusoidal torque: T L = 0.15 sin(10πt) + 0.3 N · m during 12 ≤ t ≤ 15 s. 3) Sawtooth torque: T L repeatedly and monotonically increases from 0 to 0.3 N · m and decreases to 0 N · m during 21 ≤ t ≤ 27 s. The method proposed in this article will be tested under various different working conditions. The reference speeds are 500 and 1200 r/min, respectively. As shown in Figs. 5(a)-(c) and 6(a)-(c), the presented RTLMC law exhibits a much higher steady-state accuracy under the three kinds of load torque disturbances. Figs. 5(d)-(f) and 6(d)-(f) reveal that the proposed RTLMC and FBFC have similar q-axis currents corresponding to the same working condition. The reason for this phenomenon is that the q-axis current is sensitive to value of the speed. Weak change of the q-axis current will lead to obvious change of speed. In Figs. 5(a)-(c) and 6(a)-(c), the maximum difference in speed under the action of the two controllers is about 10 r/min, which is not enough to cause a significant change in the q-axis current. In addition, our control goal is speed regulation, and the response of the q-axis current is not our focus. We conduct the same performance test under the working condition of reference speed is 800 r/min, and the obtained results are similar to the aforementioned conclusions, so we omit them here.  In summary, the experimental results verify the effectiveness of the proposed RTLMC, which can simultaneously realize the complex mixed temporal logic speed regulation task and compensate the unmatched load torque variations, while maintaining the satisfying speed tracking performance.

V. CONCLUSION
In this article, the STL specifications and unmatched disturbance rejection problems were studied simultaneously for a class of second-order motion control systems. A continuous feedback controller for atomic temporal logic formula was designed by using PPC combined with DOBC, which can maximize the robust semantics of the atomic temporal logic formula. Different from the existing control strategies for temporal logic specifications, this article developed a novel backstepping control framework that successfully fulfilled compensation for the effects caused by matched/unmatched disturbances on the basis of satisfying the STL specifications. Finally, a switched control strategy was proposed, which allowed using several PPF constraints to control systems under STL specifications consisting of such atomic temporal logic formulas. Experimental results were given to verify the feasibility and effectiveness of the presented control method.

APPENDIX A PROOF OF THEOREM 1
The whole proof is divided into two steps. To begin with, we prove that the prescribed boundary (10) is satisfied. Then, we explain how the formula φ is satisfied.