Research On Control Method For The Discrete-time Switching Systems

In control engineering, there exist control systems which often have the character-istic of mode transitions, therefore, the control problems of switching systems are takeninto consideration, its research has both theoretical significance and engineering value.In this thesis, a class of switching systems is discussed, whose discrete mode tran-sitions are caused by discrete disturbance events and the continuous dynamical systemis described by discrete-time difference equation, the state-feedback control and output-feedback control problems are investigated deeply. During the controller designs, thefactors which affect the dynamics of switching systems are considered, such as the pa-rameter uncertainty、disturbances、nonlinear dynamics、time-domain satisfaction andthe required performances. Accordingly, the feasibility of applying the proposed methodsto idle speed control problem (ISCP) is discussed.In the presence of state and control input bounds, the state-feedback stabilizing prob-lem of linear switching systems is researched. Based on the concept of positive invariance,a state-feedback controller design methodology is proposed. The basic idea of this methodis to design the state constraint set as a controlled invariant one for closed-loop systems,and guarantee the state constraint set contained in a polyhedral domain of linear behaviorwhere the control input does not saturate. Among the stabilizing controllers above, it isof great importance to select the controller which can guarantee the closed-loop switch-ing systems have the optimal convergence rate. An equivalent condition which impliesthe closed-loop systems have the optimal convergence rate is proposed, then the selectionof optimal stabilizing controller is converted to an optimization problem, which is easyto implement. The proposed controller design method in this chapter can guarantee thestability of systems, and the closed-loop switching systems have the optimal convergencerate, also the constraints on state and control input are satisfied.In practice, the control systems always suffer uncertainties, if these influences are nottaken into account, the expected control objectives cannot be achieved. For linear switch-ing systems with parameter uncertainty or continuous disturbances, the state-feedbackstabilizing problems are discussed. In the case of parameter uncertainty, the relation be-tween robust stability of systems and the existence of robust controlled invariant set is given, and the condition which can test the robust controlled invariance is converted toa problem that can be judged in finite-step calculations. Then, based on the extendedpositive invariant approach, a condition to test the robustness of a state feedback con-troller which can stabilize the nominal linear switching systems is proposed, and obtainthe robust stabilizing controller. In the presence of continuous disturbances, in order toguarantee the ISS (Input-to-State Stability) of systems, the concept of controlled D invari-ance is introduced. Taking the similar idea, a procedure which can stabilize the disturbedlinear switching systems and also satisfy the state and control input constraints, is given.The form of controller in this chapter is simple, and can be obtained from the judgment ata finite number of points or solving of a finite number of inequalities.Usually, the state of systems cannot be directly measured, then it is an importantcontrol problem to design output feedback controller by using the measured output ofsystems to guarantee the stability of closed-loop systems. For linear switching systems,taking into account the concept of the minimum dwell time, a new condition of stabilityis presented, which is the existence of δm-controlled invariant set. In the case of invariantellipsoids, existence conditions to obtain such a set are given. Then, under the assumptionthat the continuous state is unavailable for feedback, this result is used to find the observer-based output feedback controller and ordinary output feedback controller design methods.The proposed approaches in this chapter can be obtained from the feasible problem ofLMI, and easy to calculate.In practice, the nonlinear dynamics is widespread existence, it is practically impor-tant to consider the effects of nonlinear terms in the dynamical system. For switching sys-tems with Lipschitz-like nonlinear term, full-state feedback controller design、continuousstate observer design and observer-based output feedback controller design problems areconsidered. The basic idea of the proposed approaches is to construct the different types ofHybrid Lyapunov function which are not increasing to guarantee the stability of closed-loop systems. In the framework of LMI, all of those methods are given in the form offeasible problem of LMI. The results obtained in this chapter are only dependent on theLipschitz-like constant matrices without regard to the specific nonlinear forms in switch-ing systems, easy to implement.For the constrained switching systems with in-loop transitions and controlled affine-resets, in the presence of parameter variations and additive disturbances, the control prob- lem of determining the non-conservative bounds on the l∞performance of the controlledoutput is considered, and the controller design methodology which can obtain the minimall∞performance is proposed. The basic idea is to translate the required level of perfor-mance into the safety problem of systems, then the algorithm for maximal safe set andcontroller design methodology are given, together with the bisection method, the minimall∞performance can be obtained. Meanwhile, this method is applied to controller designof ISCP. For the4-cylinder4-stroke spark ignition in-line engine in idle mode, taking intoaccount the effect of disturbance torque and air temperature, a reduced switching model ofISCP is given, also, the control problem of ISCP is considered as the minimal l∞perfor-mance problem of switching systems in this chapter, then a controller design technologyof ISCP is presented. Different simulation results show that the controller of ISCP has agood control performances.