Study On Lyapunov-based Deterministic Robust Control (LDRC) Of Uncertain Mechanical Systems

In this dissertation, we focus on studying deterministic robust control design of uncertain mechan-ical systems (unconstrained and constrained) via Lyapunov stability theory.We first propose a deterministic robust control scheme for unconstrained uncertain mechanicalsystems whose uncertainty bound is known. The inertia matrix’s singularity and upper bound propertyare discussed in detail. The inertia matrix may be singular due to over-simplified modeling. A robustcontrol scheme is proposed to deal with the uncertainty in mechanical systems. It is demonstratedthat based on the non-singularity and upper bound assumptions of the inertia matrix, one can indeedutilizes the inertia matrix to construct a legitimate Lyapunov function candidate for control designand stability analysis. The control does not need to know the uncertainty information of the dynamicsystem other than its upper bound.Then, a novel optimal robust control approach of unconstrained uncertain mechanical systems(whose uncertainty bound is fuzzy) is proposed. Fuzzy set theory is used to describe the bound ofuncertainty. The desirable system performance is deterministic (assuring the bottom line) and alsofuzzy (enhancing the cost consideration). The proposed control is deterministic and is not the usual if-then rules-based. The resulting controlled system is uniformly bounded and uniformly ultimatelybounded proved via the Lyapunov minimax approach. A performance index (the combined cost,which includes average fuzzy system performance and control effort) is proposed based on the fuzzyinformation. The optimal design problem associated with the control can then be solved by minimizingthe performance index. The unique closed-form optimal gain and the cost are explicitly shown. Theresulting control design is systematic and is able to guarantee the deterministic performance as well asminimizing the cost.Fundamental equation of constrained mechanical systems is obtained based on Udwadia-Kalabaapproach. According to Udwadia-Kalaba approach, we first consider the unconstrained mechanicalsystem whose equations of motion can be written by using Newtonian or Lagrangian mechanics interms of the generalized coordinates. Then we form the second-order constraint equations of differ-ent kinds of constraints. In the end, impose the additional generalized forces of constraint obtainedfrom the second-order constraint equations upon the unconstrained system. By using this modelingmethodology for multi-body systems (constrained discrete dynamical systems, including constrainedmechanical systems), we can always derive Udwadia-Kalaba equation in the explicit analytic form.The explicit equations of motion based on Udwadia-Kalaba approach are applicable to systems with holonomic or nonholonomic constraints, with ideal or non-ideal constraint forces. It is also applicableto systems whether or not their mass matrices are singular. Unmatched ease, clarity and elegance ofUdwadia-Kalaba approach for solving multi-body system are presented.For the control part, we first consider the constrained mechanical system without uncertainty. Theissue of generation of constraint forces is considered as a control task (that is servo constraint prob-lem). Based on the obtained dynamical model, constraint-following servo control is used to realize theperformance requirement modeled as servo constraints of second-order. That is to use a set of servocontrols to generate the appropriate constraint force for the system to obey the constraints. The inversedynamics control problem based on Moore-Penrose generalized inverse, second-order constraints andconstraint violation suppression utilized in the proposed approach are explained. Furthermore, thisproposed approach is applicable to not only fully actuated but also under-actuated and redundantlyactuated mechanical systems. Two-mass spring system and coordinated robot system are presented asexamples for illustration.We then consider the constrained mechanical system with uncertainty. The uncertainty (possiblyfast time-varying) in mechanical systems is bounded. The performance requirement is also modeledas servo constraints in second-order form. A model-based robust control is presented to approximatelyfollow the servo constraints (With the uncertainty in presence and no restrictions on the initial con-dition, it is only reasonable to expect the system to follow the prescribed constraints approximately).The proposed robust control is motivated by the perfect constraint following control design which isbased on the D’Alembert’s principle (the Nature’s action). The robust servo control is used to gen-erate appropriate constraint force for the system to approximately obey the constraints. The uniformboundedness and uniform ultimate boundedness of the tracking error are guaranteed, regardless of theuncertainty. State transformation and force feedback are not needed. Coordinated robot system ispresented as examples for illustration.Simulation results are used to corroborate the theoretical findings of this work.