formation for n3(t) : .... systems: a design via LMIs" , IMA Journal of Math- ... "Memo- ryless Stabilization of Uncertain Dynamic Delay Sys- tems: Riccati Equation ...
2005 lnternational Conference on Control and Automation (ICCA2005) June 27-29,2005, Budapest, Hungary
Block Predictor Sliding Mode Control of Linear Time Delay Sytems O. Espinosa-Guerra*, Alexander G. Loukianov* and B. Castillo-Toledo* * Deparment of Electrical Engineering, CINVESTAV IPN, Apartado Postal 31-438, Guadalajara, Jalisco, 44550, México. email:{louk, oespinos, toledo)@gdl.cinvestav.rnx. Abstract This paper applies the block control method t o form a decomposed discontinuous control law suitable for multivariable linear time-delay systems. A block controllable form is introduced and a non-singular transformation that reduces the system to this form is proposed. A block delay compensation algorithm which gives a sliding manifold is derived. An example of the application of the proposed control strategy is illustrated.
1 Introduction The feedback stabilization of time-delay systems remains is one of most interest problems in control theory because many industrial proceses modelled by delay differential equations. This problem has been extensively studied and several controllers and stability criteria based on optimal control method see [13], [12], [l], including H , and LMI approaches [5], [8],or averaging theory [6], have been proposed. The common feature of the referred papers is that their derivations are based on analysis of full order system. To decompose the design procedure and introduce a robustness property the sliding mode technique have beeri applied [3], [4], 121. In this paper, to stabilize linear time invariant systems with delayed state and input, we use the block control pririciple which is fruitful and relatively simple, especially when dealing with multivariable systems because the control problem is decomposed into a number of simpler sub-problems of lower diinensions. In order to achieve this, a special state representation must be used which will be referred to as the Block Controllable f o m (or BC-form), consisting of a set of controlled blocks. This approach has successfully been employed to stabilize linear time delayed systerns [9]. Here, the possibility of applying the same method to design a predictor based sliding mode control, is investigated. Note that a sliding rnode predictor controller was designed [ll]for linear systems with delay only in the control input. We
consider a linear systein with delay in the both state and control variables.
2 Block Controllable Form for Systems
with Delay Consider a linear time-delay system described by the following state equation
where x E R n , u E Rm and A, C, B and D are matrices of appropriate dimensions, and x(t) = O
where the equivalent control u,, is calculated from u = 0 as
with a l l = - a, bl = ak, a31 = -w2, a 3 = ~ -2
