Structured Optimal Control Method Based On Parameter Space Optimization

The integrated optimization design of mechanical,electrical and control module parameters can accelerate the design iteration process of complex systems and achieve high system performance.By lumping together the mechanical,electrical,and control parameters into the feedback gain matrix of the dynamic system and using the optimal control theory to solve it,the optimization speed is expected to be improved and the demand for the fast reconfiguration of the flexible manufacturing line is satisfied.However,in complex systems,due to the influence of mechanical structure,driving scheme and controller architecture,the feedback gain matrix of the system has structural constraints and cannot be solved by using the algebraic Riccati equation（ARE）.The controller design problem with a structurally constrained feedback gain matrix is usually called structured optimal control problem.By establishing the optimization goal and the structural constraints of the feedback gain matrix,etc.from state space to parameter space,and using the highly efficient method based on convex programming techniques,the global optimal solution is expected to be obtained.Also,the closed-loop stability of the system dynamics is ensured in the presence of convex-bounded model uncertainties.However,at present,such a method only copes with limited types of structural constraints,i.e.,either some of the entries in the feedback matrix are zero or some others in the same row are equal or opposite to each other.Also,the solution has weak robustness and the numerical process is computationally inefficient.These issues hinder its applications in the integrated optimal design of large and complex electromechanical systems.In view of the problems illustrated above,this thesis carries out the following several aspects of work:First,the transformation relationship is revealed between the dynamic system with constrained feedback gain matrix and the unconstrained decentralized feedback system.The mapping rules are established for theH₂ optimization goal,constraints and dynamic system from state space to parameter space.Then the optimal control parameters can be solved for the feedback gain matrix with the arbitrary linear equality constraints between the intra-row and intra-column entries.The dominant pole constraint is introduced into the optimal control problem to prevent the closed-loop system from falling into marginal stability when the model parameters are at the boundaries of the uncertainty domain.The logic of constraints generation for iterative dual-simplex linear programming（LP）algorithms is improved,so that multiple cutting planes can be generated in a single iteration.This improves the computing efficiency.Then,the structural constraints are extended to a more general case which allows the existence of the linear equality constraints between cross-row and cross-column entries in the feedback matrix.Based on some properties of the Kronecker product,we propose a bilinear factorization method of the feedback gain matrix.This factorization has the form of a block-diagonal feedback gain matrix in the middle and two constant matrices on two sides.By this method,only the block-diagonal structure and the constraints of the additional identical diagonal blocks remain after applying the factorization to the feedback gain matrix.For structured H₂ optimal control problems,we define the explicit mapping of the optimization goal,constraints and system dynamics from state space to parameter space.Then we simplify the original optimization problem to a convex optimization problem which is iteratively solved by using the cutting plane algorithm.Finally,we extend the above structured optimal control design method toH₂,H_∞,and mixedH₂/H_∞robust controller synthesis for precision trajectory tracking.Under the two-degree-of-freedom control architecture of feedforward control and feedback control,we discuss the structural feedback constraints when considering the dynamics of the trajectory.We further point out that the current mapping method for sparse feedback structure cannot achieve a higher performance in tracking a specific trajectory due to the conservativeness of dealing with the sparse structure of the feedback gain matrix.A less-conservative constraint with the coupling of trajectory dynamics and system dynamics is proposed,which lays the foundation for the subsequently structured optimal control design in the parameter space.Several examples are introduced to verify the effectiveness of the proposed structured optimal control method such as the integrated optimal design of mechanical and control parameters of electromechanical systems,robust tracking control of the coarse/fine stages and multi-agent tracking systems.Finally,the deficiencies of the current research are pointed out and the directions for future work are suggested.