The Research Of Control Problem On Linear Constrained System

Engineering systems usually run with fixed bounds, and control actions must satisfy strict limitations. Therefore, constrained control problem emerges. Linear constrained system is simple in form, approximates the facts and has tractability in theory and practicability in project. Linear constrained system can describe many objects in many fileds. When constraints are ineluctable factors, we must deal with constraints while designing the controller of systems, otherwise, the expected index could not be obtained and bring tragedies sometimes. This thesis researches stabling problem of linear constrained system.The positively invariant sets are special form of convex polyhedral sets, which connect constrained conditions and play an important role in design of controller. State feedback unincreases new state, unchanges controllability and unimpacts input matrix. The control problem of linear systems with constraints of control, state and initial state is studied. Using positively invariant sets of dynamic system, state feedback controller is designed so that initial states transfer asymptotically while all constraints are satisfied. First, the problem of the existence of control law is presented and stability problem of constrained system is converted to minimization problem of linear objective function by using positively invariant sets and optimization theory. Then, the nonlinear optimization problem is solved and the control gain of linear state feedback control law is obtained. The approach has two virtues: (1) Because control gain minimize the objective function and objective function is in inverse proportion to the rate of convergence o, so states converge to equilibrium at the fastest rate with control law. (2) Control gain can beobtained by solving optimization problem, and optimization problem has no relation with current state of system, therefore, synthesis of control law has separability. Namely, we can get control gain off line and measure the state at every time. The product of control gain and state is current control. This separability is expected on engineering. Finally, simulation results show that control law obtained is effective.