On the Gain Margin Improvement Using Dynamic Compensation Based on Generalized Sampled-Data Hold Functions

This paper shows that the use of dynamic compensation based on generalized sampled-data hold hmdious (GSHF) can arbitrarily improve the gain margin for continuous-time mnminimum phase linear systems The GSHF compensator is a particular periodic digital contrdler mu& simpler than that used in [4]. The effect of sampling period on the gain margin is analyzed. Furthermore, it is proved that under a mild condition, the gain margin improvement can be achieved without forcing the sampling period small. An important advantage of periodic compensation over LTI compensation is found to be the capab'ity of reducing conflict between gain margin and sensitivity, which always exists for a nonminimum phase plant as far as LTI compensation is concerned [141. Generally speaking, there are two common kinds of periodic digital controllers for continuous-time linear time-invariant (LTI) plants. A controller of one kind (i.e., a dynamical system with periodically time varying elements) is composed of a sampler, a periodic discrete-time dynamic component, and a zero-order hold [7] while a controller of the other kind is composed of a sampler, an LTI compensator, and a periodic control gain known as a generalized sampled-data hold function (GSHF) [S]. We might call the former a conventional periodic digital controller and the latter a GSHF dynamic compensator. Their common feature is the hybrid nature of a continuous-time component and a discrete-time component. The most significant difference between the two configurations is that periodicity of a conventional periodic controller occurs in the dynamic component while periodicity of a GSHF compensator occurs only in the GSHF gain with any dynamic components time-invariant. Evidently, this difference implies that a GSHF dynamic compensator may be more easily implemented in practice than a conventional periodic digital controller. One of the known advantages of periodic controllers over LTI controllers is their capability of improving arbitrarily the gain margin for an LTI plant in certain cases; see e.g., [7], [9]. Recently, Francis and Georgiou [4] showed that for a discrete-time LTI plant, LTI dynamic pre-compensation with decimation of the plant output (which is equivalent to use of a particular form of linear periodic dynamic compensator) can arbitrarily place nonzero zeros of the resulting system. Using this key idea, they generalized the gain margin result in [7] to the multivariable case. More specifically, the gain margin can be arbitrarily assigned likewise for a multivariable continuous-time LTI plant by way of discretizing the plant with a sufficiently small sampling time and suitable choice of digital periodic controller. According to their method, the design procedure consists of i) discretizing the plant, ii) designing an LTI dynamic fonvardcompensator for the discretized plant with output decimation (this positions zeros), and iii) designing an LTI feedback compensator Manuscript received July, 1993; revised December 30, 1993. W.-Y. Yan is with the Department of Mathematics, University of Western Australia, Nedlands, W.A. 6009, Australia. B. D. 0. Anderson and R. R. Bihnead are with the Department of Systems Engineering, Research School of Physical Sciences and Engineering, Australian National University, GPO Box 4, Canberra, ACT 2601, Australia. IEEE Log Number 9405653. (this positions poles). Altogether, this actually leads to a conventional periodic digital controller. From this procedure, it is not hard to observe that the order of such a controller may be very high because of the introduction of precompensation. Other disadvantages, as mentioned in [4], are that the sampling time may have to be very small to permit an increase in the gain margin and the feedback system may become sensitive to variation in parameters other than the gain. As is well known, the idea of a GSHF function can be powerfully used for many linear control system problems; see [2], [S]. Particularly in [S ] , Kabamba exhibited some advantages of GSHF nondynamic compensation over LTI compensation. Naturally, the use of GSHF dynamic compensation might be expected to achieve even more profitable objectives. One of the main purposes of this paper is to reveal one of the objectives, the gain margin improvement, and to examine the effect of sampling time on the gain margin. Even though periodic compensation can arbitrarily improve the gain m e n , this compensation cannot improve the minimal sensitivity for an LTI plant, a fact which has been shown by Khargonekar et al. [7] as well as by Feintuch and Francis [3]. Recently, in [14], we revealed that there always exists conflict between gain margin and sensitivity for a nonminimum phase LTI plant using LTI compensation. For instance, gain margin maximization will lead to an arbitrarily large sensitivity. Thus, an interesting question naturally arises as to whether periodic compensation can overcome or reduce this compromise associated with LTI compensation. Our partial answer to this question in this paper shows that the use of periodic compensation cannot only arbitrarily improve the gain margin but also keep the sensitivity bounded at the same time. In the next section, a GSHF dynamic compensator is fonnulated and a stabilizability condition for a continuous-time LTI plant with the compensator is established. In Section III, we derive an explicit formula for the maximal achievable gain margin of a singleinputlsingle-output (SISO) plant by a GSHF compensator, analyze the effect of sampling time on the maximal gain margin, and show that an arbitrarily large gain margin can be achieved by a GSHF compensator with a sufficiently small sampling period. Section IV discusses the multivariable case. We exhibit the capability of reducing conflict between gain margin and sensitivity by using periodic compensation in Section V. An example is given in Section VI. Consider a continuous-time LTI plant P ( s ) with a minimal statespace model x( t ) = Ax(t) + Bu(t) y ( t ) = C x ( t ) + Du(t) where x(t) E Rn is the state, u( t ) E Rm is the control, y(t) E Rr is the output, and A, B , C , D are real matrices of appropriate dimensions. The GSHF dynamic compensator is defined to be the following control law composed of an LTI compensator and a GSHF feedback ~ ( t ) = F(t)vk, fort E [kT, (k + 1)T) , k = 0, 1 , 2 , . . . (2.5) 0018-9286/[email protected] O 1994 IEEE 2348 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39. NO. I I , NOVEMBER 1994 where T > 0 is the sampling period, A,. B,. C,. and D, are constant matrices of appropriate dimensions, and F ( t ) is a T-periodic integrable and bounded hold function matrix of an appropriate dimension. The following equation for the unknown F ( t ) with G a given constant matrix plays an important role in the design of the GSHF control law. The properties with respect to this equation are summarized in the following result. Lemma 2.1: Let (A . B ) be controllable, G be given and 1 . B. T = exp [ d ( T t ) ] B B r exp [.-IT(T t ) ] dt. (2.7) IT