Incremental Model Predictive Control Exploiting Time-Delay Estimation for a Robot Manipulator

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Due to uncertainties and/or disturbances, the Lyapunov stability established for the nominal system is not simply transferable, and input-to-state stability is required to be considered.

“Due to uncertainties and/or disturbances, the Lyapunov stability that was established for the nominal system is not simply transferable [17], [18], and input-to-state stability (ISS), also known as practical Lyapunov stability, is required to be considered.”

• cites [17] sourcePartly supported

  Section 3.2 (p. 4) establishes a Lyapunov function for the NMPC closed loop and concludes asymptotic stability under its stated assumptions. The passages found do not address whether that stability transfers to systems with uncertainties or disturbances, or say that input-to-state stability must be considered. (LLM, conf=0.83)

  Experimental — accuracy not yet measured
• cites [18] sourceNot checked — no full text of the source

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In existing MPC without terminal ingredients, prediction is based on a nominal model requiring identification of the nominal model with high precision, whereas the proposed IMPC employs the incremental model based on TDE and thus it is not necessary to identify the model of f(x) and g(x).

“In existing MPC without terminal ingredients, comparing [17] and [18], the prediction is based on a nominal model. This requires to first identify the nominal model of the plant with high precision, whereas the proposed IMPC approach employs the incremental model based on TDE and, thus, it is not necessary to identify the model of f ( x ) and g ( x ) in (2b).”

• cites [17] sourcePartly supported

  Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured
• cites [18] sourceNot checked — no full text of the source

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Experimental results show that the system can achieve optimal control performance while guaranteeing that input and state constraints are not violated.

“Experimental results show that the system can achieve optimal control performance while guaranteeing that input and state constraints are not violated.”

The quote only partly states this claim.

If X ∈ D, Y ∈ ̄X, and X - Y_Q^2 ≤ cd, then the difference between VN(Y,k) and VN(X,k) is upper bounded with a constant K > 0.

“Lemma 5 (Continuity of Value Function): If X ∈ D , Y ∈ ¯ X , and X -Y 2 Q ≤ cd , then”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

For A in (13), its eigenvalue λ(A) > 1.

“Unfortunately, for A in (13), its eigenvalue λ( A ) > 1.”

The quote only partly states this claim.

Let Assumptions 1-3 hold and ̄∗ satisfy the stated condition; considering a reachable reference trajectory, there exists N1 > 0 such that for all N > N1, VN(X(k),k) with initial value VN(X(k0),k0) ≤ Vmax is an ISS Lyapunov function.

“Theorem 1: Let Assumptions 1-3 hold and suppose that ¯ ∗ satisfies”

• AssumptionThe authors state this as an assumption: The TDE error satisfies assumptions (29) and (30).
• AssumptionThe authors state this as an assumption: For X ∈ D and Y ∈ R^4n, Y - X_Q^2 ≤ cd implies that for all i ∈ [0,N-1], Y_{k+i|k} ∈ ¯X and u_{Y,k+i|k} ∈ U, where Y_{k+i|k} and u_{Y,k+i|k} are predictions generated by the locally optimal incremental controller.

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For the locally optimal incremental controller (19), upper bounds for the tracking errors and for the control signals in the prediction horizon are obtained.

“For the locally optimal incremental controller (19), upper bounds for the tracking errors are obtained (cf. (58) in Appendix A), and for the control signals in the prediction horizon”

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Assuming ̄∗ ≤ √(cd/λmax(Q)), if X(k) ∈ D and N ≥ N1, then X∗k+1|k ∈ D and X(k+1) ∈ ̄X.

“Lemma 6: Assuming that ¯ ∗ ≤ √ ( cd /λ max ( Q )) , if X ( k ) ∈ D and N ≥ N 1, then X ∗ k + 1 | k ∈ D and X ( k + 1 ) ∈ ¯ X .”

• AssumptionThe authors state this as an assumption: For X ∈ D and Y ∈ R^4n, Y - X_Q^2 ≤ cd implies that for all i ∈ [0,N-1], Y_{k+i|k} ∈ ¯X and u_{Y,k+i|k} ∈ U, where Y_{k+i|k} and u_{Y,k+i|k} are predictions generated by the locally optimal incremental controller.

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In Scenario 2, input constraints are not violated and the system is still stable, although the tracking errors are larger than in Scenario 1.

“As shown in Fig. 6, input constraints are not violated and the system is still stable, although the tracking errors are larger than that in Scenario 1.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Assuming E(k)^2_Q <= c, the theoretical minimal prediction horizon is N = 5.02 x 10^12.

“Assuming that E ( k ) 2 Q ≤ c , we obtain the theoretical minimal prediction horizon N = 5 . 02 × 10 12 .”

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In Scenario 4, for joints 1 and 2, there are no significant differences in tracking errors and tracking velocities between Scenarios 1 and 4.

“For joints 1 and 2, there are no significant differences in tracking errors and tracking velocities between Scenarios 1 and 4.”

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Although ISS of IMPC is not theoretically proven for unreachable reference trajectories, the reference trajectory is tracked as close as possible and the closed-loop system is still stable in Scenarios 2-5.

“Although ISS of IMPC is not theoretically proven for unreachable reference trajectories, the reference trajectory is tracked as close as possible and the closed-loop system is still stable (as shown in Scenarios 2-5).”

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In Scenario 3, for joints 1 and 2 the reference is reachable and the tracking errors decrease as the prediction horizon N increases.

“For joints 1 and 2, the reference is reachable and tracking errors decrease with increasing the horizon N .”

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Although ISS for unreachable reference trajectories is not considered in the presented theory, the system outputs stay close to the reference signals.

“Although ISS for unreachable reference trajectories is not considered in the presented theory, system outputs stay close to reference signals.”

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For unreachable trajectories, the iteration number of the qpOASES solver increases to keep constraint satisfaction, which also increases the computing time.

“However, for unreachable trajectories, an iterative number of the qpOASES solver increases to keep constraints satisfaction, and it will also increase the computing time at the same time.”

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Most of the computing time for each prediction horizon is smaller than 0.2 ms, which verifies real-time control capability.

“As shown in Fig. 5, most of the computing time for each prediction horizon is smaller than 0.2 ms, which verifies real-time control capability.”

The quote only partly states this claim.

For the unreachable reference trajectory in Scenario 6, as the horizon N increases the output is closer to the reference signal.

“as the horizon N increases, the output is closer to the reference signal.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In Scenario 5, joint 3 operates stably similar to Scenario 3.

“Joint 3 operates stably similar to Scenario 3.”

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For Ts = 0.02 s, the tracking error trajectory starts to oscillate because a larger TDE error affects the solutions of the constrained OCP (18) and the system becomes unstable.

“For Ts = 0.02 s, the tracking error trajectory starts to oscillate because a larger TDE error affects solutions of the constrained OCP (18) and the system becomes unstable.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In Scenario 5, for joint 2, there are no significant differences.

“For joint 2, there are no significant differences.”

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The tracking error increases as the sampling period increases because the TDE error increases with increasing sampling period.

“The tracking error increases as the sampling period increases since the TDE error increases with increasing the sampling period as analyzed in Section IV-B.”

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The proposed incremental model predictive control (IMPC) strategy allows for constrained control of a robot manipulator while deriving the incremental model without a concrete mathematical system model.

“This article proposes a new incremental model predictive control (IMPC) strategy, which allows for constrained control of a robot manipulator, while the resulting incremental model is derived without a concrete mathematical system model.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Nonlinear control approaches such as computed torque or backstepping are able to deliver high-performance controllers for robot manipulators.

“Nonlinear control approaches, such as computed torque [4] or backstepping [5], are able to deliver high-performance controllers for robot manipulators.”

• cites [4] sourceNot checked — no full text of the source
• cites [5] sourceNot checked — no full text of the source

Classified as probably not a substantial claim. May combine two claims.

Intelligent control techniques such as fuzzy logic systems and neural networks are proposed to approximate unknown or uncertain system dynamics, but they focus on robustness rather than optimal tracking performance, and input and state constraints are usually not of interest.

“To alleviate the need of concrete mathematical modeling and parameter identification, intelligent control techniques, such as fuzzy logic systems [10] and neural networks (NNs) [11], are proposed to approximate unknown/uncertain system dynamics. Yet, these approaches also pay more attention to robustness, not to optimal tracking performance. In addition, input and state constraints are usually not”

• cites [10] sourceNot checked — no full text of the source
• cites [11] sourceNot checked — no full text of the source

Classified as probably not a substantial claim. May combine two claims.

For MPC, the constrained optimal control problem is solved using state predictions generated by the nominal mathematical model in a horizon, where the difference between nominal and real models due to uncertainties and disturbances may degrade tracking performance.

“For MPC, the constrained optimal control problem (OCP) is solved using the state predictions generated by the nominal mathematical model in a horizon [14]-[18], where the difference between the nominal and real mathematical models due to uncertainties and disturbances may degrade tracking performance [19].”

• cites [14]-[18]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [19] sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing).

Model predictive control is an optimization-based control addressing optimal tracking control performance and input and state constraints.

“Model predictive control (MPC) [12], or receding horizon control [13], is an optimization-based control addressing optimal tracking control performance and input and state constraints, such as saturation of control inputs and physical limits of workspace and speed.”

• cites [12] sourceNot checked — no full text of the source
• cites [13] sourceNot checked — no full text of the source

Classified as probably not a substantial claim. May combine two claims.

Model uncertainties and disturbances increase the computational complexity of MPC and make stability analysis more challenging.

“Model uncertainties and disturbances not only increase the computational complexity of MPC, but they also make stability analysis more challenging.”

May combine two claims.

In [30] and [31], ISS of MPC with terminal ingredients was analyzed using a feasible control sequence, but the derived cumulative error bound increases with increasing prediction horizon, resulting in an overconservative cumulative error bound.

“In [30] and [31], ISS of MPC with terminal ingredients was analyzed where a feasible control sequence, which guarantees that input and state constraints are not violated, was employed. However, the derived cumulative error bound increases with increasing the prediction horizon. It results in an overconservative cumulative error bound”

• cites [30] sourceNot checked — no full text of the source
• cites [31] sourceNot checked — no full text of the source

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Using continuity of the value function, ISS is verified and an overconservative cumulative error bound is avoided.

“Using continuity of the value function, ISS is verified and overconservative cumulative error bound is avoided.”

Classified as probably not a substantial claim.

The TDE error is bounded if a TDE-based controller is employed.

“In [36]-[40], [44], and [45], the TDE error is verified to be bounded if a TDE-based controller is employed.”

• cites [36]-[40]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [44] sourceNot checked — no full text of the source
• cites [45] sourceNot checked — no full text of the source

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The TDE-based incremental controller is developed in the framework of MPC without terminal ingredients, and input and state constraints are formulated as inequality constraints, achieving optimal control performance while considering constraints, which has not been considered in existing TDE-based controllers.

“Optimal Performance for TDE-Based Control: The TDE-based incremental controller is developed in the framework of MPC without terminal ingredients, and input and state constraints are formulated as inequality constraints. Thus, optimal control performance is achieved, while input and state constraints are considered, which has not been considered in existing TDE-based controllers [36]-[40].”

• cites [36]-[40]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

Other robust MPC schemes including min-max optimization and sliding mode techniques improve robustness without significant increase of complexities.

“Other robust MPC schemes, including min-max optimization [25], [26] and sliding mode techniques [27], [28], are also employed to improve robustness without significant increase of complexities.”

• cites [25] sourceNot checked — no full text of the source
• cites [26] sourceNot checked — no full text of the source
• cites [27] sourceNot checked — no full text of the source

Classified as probably not a substantial claim.

TDE is a model-free method that uses time-delayed input and output signals to estimate partial dynamics of the system without concrete mathematical model, laborious parameter identification, and linearization around equilibrium points.

“TDE is a model-free method, which uses time-delayed input and output signals to estimate partial dynamics of the system without concrete mathematical model, laborious parameter identification, and linearization around equilibrium points of the system [38]-[40].”

• cites [38]-[40]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

In practice, increasing prediction horizon appropriately will decrease tracking errors.

“However, in practice, increasing prediction horizon appropriately will decrease tracking errors [32].”

• cites [32] sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S4 (based).

The linear system (A,B) is controllable.

“Note that the linear system ( A , B ) is controllable.”

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Different from conventional linearization methods such as Taylor series and Carleman approaches, the linear system (13) is obtained without a concrete mathematical model and is effective in the whole set D_x, not only locally around equilibrium points.

“Different from conventional linearization methods, such as Taylor series [29], [47] and Carleman approaches [48], the linear system (13) is obtained without concrete mathematical model. Besides, it is effective in the whole set D x , not only locally around equilibrium points.”

• cites [29] sourceNot checked — no full text of the source
• cites [47] sourceNot checked — no full text of the source
• cites [48] sourceNot checked — no full text of the source

May combine two claims.

Different from existing ISS analyses for MPC, continuity of the value function is used to derive the upper bound of the difference between two value functions, and the resulting cumulative error bound is not overconservative.

“Different from existing ISS analyses for MPC, here, continuity of the value function is used to derive the upper bound of the difference between two value functions, and the resulting cumulative error bound is not overconservative.”

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It is theoretically inferred from the ISS and the cumulative error bound that increasing the prediction horizon enlarges the region of attraction and at the same time decreases tracking errors.

“It is theoretically inferred from this ISS and the cumulative error bound that increasing the prediction horizon enlarges the region of attraction and, at the same time, decreases tracking errors.”

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The cost function J_N is convex because of its quadratic form; (17b) is an affine system and ̄X and U are convex sets; therefore OCP (18) is a convex optimization problem, and a unique optimal solution exists for nonempty admissible sets.

“Obviously, the cost function JN ( X ( k ), ¯ u ( k ), k ) is a convex function because of its quadratic form. Besides, (17b) is an affine system and ¯ X and U are convex sets. Therefore, OCP (18) is a convex optimization problem, and a unique optimal solution exists for nonempty admissible sets.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

MPC formulated with an incremental model generates state predictions by using both present and previous states, but since the incremental system is obtained from linearization of the nonlinear system, the controller is effective only locally around equilibrium points.

“Alternatively, MPC is formulated with an incremental model, which generates the state predictions by using both present and previous states [29]. However, since the incremental system is obtained from linearization of the nonlinear system, the controller is effective only locally around equilibrium points.”

• cites [29] sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

A robust MPC was proposed for a helicopter where an extended high-gain observer estimates model uncertainties and disturbances, but the nominal mathematical model is required.

“A robust MPC was proposed for a helicopter in [20] where an extended high-gain observer estimates model uncertainties and disturbances. However, the nominal mathematical model is required.”

• cites [20] sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing).

For MPC without terminal ingredients, the ISS property was investigated in [33] using time-variant sets of admissible predicted states.

“For MPC without terminal ingredients, the ISS property was investigated in [33], where time-variant sets of admissible predicted states are used.”

• cites [33] sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

A data-driven MPC uses a nonparametric machine learning technique to estimate a prediction model, but it further increases the computational complexity of MPC.

“A data-driven MPC was developed in [24], where a nonparametric machine learning technique is used to estimate a prediction model. Unfortunately, learning and data-driven techniques further increase the computational complexity of MPC.”

• cites [24] sourceNot checked — no full text of the source

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The TDE error is inevitable and stability will be affected by this error.

“However, the TDE error is inevitable and stability will be affected by this error.”

May combine two claims.

Different from existing MPC methods, the concrete mathematical model of the robot manipulator is not required because the continuous-time nonlinear system model is approximated by an incremental system using TDE and state predictions are generated using the linear discretized incremental system.

“No Concrete Mathematical Model Required: Different from existing MPC methods, the concrete mathematical model of the robot manipulator is not required. This is because the continuous-time nonlinear system model is approximated by an incremental system using TDE, and the state predictions are generated using the linear discretized incremental system.”

Probably not a substantial claim: may not pass S3 (load-bearing).

A sampling period of 0.001 s is commonly used in robotic systems.

“In our experiments, 0.001 s is used because of the setting of our experimental system, which is also commonly used in robotic systems, such as [9], [39]-[42], [45], and [53].”

• cites [9] sourceNot checked — no full text of the source
• cites [39]-[42]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [45] sourceNot checked — no full text of the source

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If the sampling period increases, the tracking error increases and the closed-loop system eventually becomes unstable.

“If the sampling period increases, the tracking error increases, and finally, the closed-loop system becomes unstable.”

May combine two claims.

The proposed IMPC achieves optimal control performance while guaranteeing that input and state constraints are not violated, without a concrete mathematical model of the robot manipulator.

“The results demonstrate the efficacy of the proposed IMPC in realizing optimal control performance while guaranteeing that input and state constraints are not violated, without concrete mathematical model of the robot manipulator.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Given the reference signals varying flatly lie strictly in the state set of Scenario 1, reference signals can be regarded as reachable reference trajectories in Scenario 1.

“Note that given the reference signals varying flatly lie strictly in the state set of Scenario 1, reference signals can be regarded as reachable reference trajectories in Scenario 1.”

Probably not a substantial claim: may not pass S1 (asserted), S4 (based).

The uncertain TDE error has to be addressed in order to guarantee strict state constraint satisfaction.

“The uncertain TDE error has to be addressed to guarantee strict state constraint satisfaction”

May combine two claims.

Due to the TDE technique, the nonlinear system is approximated by a linear system (13), and the constrained OCP (18) is a quadratic-programming problem.

“Due to the TDE technique, the nonlinear system is approximated by a linear system (13), and then, the constrained OCP (18) is a quadratic-programming (QP) problem.”

Classified as probably not a substantial claim.

A too large prediction horizon causes heavier computation such that the real-time control capability will be affected.

“However, a too large prediction horizon causes heavier computation such that the real-time control capability will be affected.”

May combine two claims.

The output approaches the reference as N increases because the region of attraction increases and the tracking error decreases with larger N.

“This is because the region of attraction increases and the tracking error decreases with larger N , as analyzed in Remarks 7 and 8.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The identified mathematical model is not used for the proposed IMPC.

“Nevertheless, note that the identified mathematical model is not used for the proposed IMPC.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

During implementation, one does not need to justify whether the reference trajectory is reachable or unreachable; only the constrained OCP (18) needs to be solved.

“During the implementation, one does not need to justify whether the reference trajectory is reachable or unreachable. We only need to solve the constrained OCP (18).”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

Generally, large ̄g results in large TDE errors, while small ̄g causes noisy responses.

“Generally, large ¯ g results in large TDE errors, while small ¯ g causes noisy responses.”