計畫編號：NSC88-2213-E032-027研究期間：199808~199907研究經費：265,000[[sponsorship]]行政院國家科學委員

蕭照焜

計畫編號：NSC88-2213-E032-027研究期間：199808~199907研究經費：265,000行政院國家科學委員

Tamkang University Institutional Repository

                                                                     1 	 Coprime Factors, Linear Matrix Inequalities, and Low-Order Controller Design    !""#$!%!&$"!&"'()*+ ' ,  - # ./   , ' - $# .0123456789:;<;=>9?   		 !"#$%& outer '(#)*+,- zero-placement )./0-convex)1 !23456789:!79;<= pole-placement)>6?@)& H∞  AB,5/0  CDEF"#G9)outer '()23456))  Abstract This paper develops a low-order controller  design method for linear discrete time-invariant  single-input, single-output systems. Using the coprime factors and the properties of discrete outer functions, the low-order controller design becomes a zero-placement problem which is convex. The solutions are obtained using the linear matrix inequality techniques. The proposed design method is used to solve for  several optimal control problems, such as pole-placement design, model-matching design, and H∞ optimization problems.  Keywords - coprime factorization, discrete outer functions, linear matrix inequalities, low-order  controller design, robust control design  1. Introduction One o f the most important design issues for many practical control problems is the design of low-order stabilizing controller. The design of  low-order controlle rs to optimize certain H2, H∞ and pole-placement involves a biaf fine matrix inequality (BMI) which is a non-convex programming problem and cannot be solved in polynomial time. As a result, several researchers [1],[4],[5],[3],[6] have shown that low-order  controllers can be obtained by solving iteratively linear matrix inequality (LMI) subproblems, which are convex and can readily be solved using existing convex optimization software such as [2]. These approach include the alternating projection method [4], the rank condition minimization method [5], and the successive substitution method in [3],[6]. In [7], noniterative schemes for designing low-order controllers, for continuous single-input single-output (SISO) systems, to optimize certain per formance indices were developed. The key step of the method proposed in [7] is the use of coprime factors such that  based on the strictly positive real functions, feedback stabilization using low-order  controllers becomes a zero-placement problem which is convex. The design does not involve any iterations so that no convergence consideration is needed.  In this paper, we will focus on the development o f noniterative schemes for  designing low-order controllers for discrete-time systems. We will only address the discrete linear  time-invariant SISO systems in this paper. We establish a suf f icient condition which guarantees a discrete trans fer function being outer. That is, the zeros are all inside the unit circle of the z -plane. Using the coprime factors and the properties of discrete outer functions, we formulate the low-order controlle r design problem as a zero-placement problem which is convex. The solutions are obtained using the LMI techniques. The proposed design method is used to solve for several optimal control problems, such as pole-placement design, model matching design, and H∞ optimization problems. The paper is organized as follows. In Section 2 we formulate the design o f low-order  stabilizing controller as an LMI feasibility problem using the coprime factors and the properties o f discrete outer function. Section 3 addresses the formulation of pole-placement design. Section 4 discusses the design of model-matching problems. The design of H∞  suboptimal control problems is presented in Section 5.                                                                      2  2. Design of Low-Order Stabilizing Controllers Consider a discrete linear time-invariant  SISO system G(z) with the minimum state-space realization )()()1( kBukAxkx +=+    (1a) )()( kCxky =                 (1b) where x is the n-dimensional state variable vector, u is the control input variable, and y is the measured output variable. We formulate a low-order controller design problem as one o f  finding an nc-dimensional controller yzKu )(−=                 (2) with nc ≤ n-1, to place the n + nc poles of the closed-loop system inside the unit circle of the z -plane, such that the closed-loop system is stable. We per form a coprime factorization o f  G(z) to obtain )()()(zDzNzGgg=                  (3) where )()()(zdzdzDfgng=, )()()(zdznzNfgng =         (4) are stable, that is, the roots of df(z)=0  are  within the unit circle of the z -plane. The coe f ficient of nz  in df(z) is set to 1. In particular, the roots of df(z)=0 are selected to be the desired closed-loop poles o f the system as o f  all the states are available for feedback control. Similarly, we per form a coprime factorization o f  the controller K(z) to obtain  )()()(zDzNzKcc=                  (5) where )()()(zdznzNccnc= , )()()(zdzdzDccnc=      (6) are stable, that is the roots of dc(z)=0 are within the unit circle of the z -plane. We will also set the coe f ficient o f cnz  in dc(z) to be 1. In particular, the roots o f dc(z)=0  are selected to be the desired poles of the closed-loop system due to the controller. Note that the controller model (2) will not be known a–priori. However, we can still speci fy dc(z) based on control  bandwidth specifications.    Let the numerators of the controller coprime factors (6) have the polynomial form  011 ...)( azazaznccccnnnncn+++= −− (7) 011)( bzbzbzd ccccnnnncn+++= −−L  (8)         We de fine the vectors a and b to contain the polynomial coefficients ],,,,[01aaaacc nnL−= ],,,[01bbbbcc nnL−=  (9) where .1=cnb     Consider the closed-loop regulation system in Figure 1. The closed-loop transfer  function T(z) from the command r  to the output y  is  )()()()()()()(zNzNzDzDzNzNzTcgcgcg+=       (10) We denote the denominator of )(zT  as )()()()()( zNzNzDzDzQcgcg+=        (11) )()()()()()(zdzdznznzdzdcfcngncngn+=  The following result shows the dependence of a  state-space realization o f Q(z) on the coe f ficients a and b. Lemma 1. The function Q(z) (11) has a state-space realization )()1()1( kuBkxAkxqqqqq++=+     (12a) qqqquxbaCky += ),()(              (12b) where xq is an )( cnn +  dimensional state vector, uq and yq are input and output variables, respectively, and Aq is Hurwitz. Furthermore, the state matrices have the following properties: (i) the pair (Aq,Bq) is controllable and is independent of a and b. (ii) Cq(a,b)=aS+bR is a linear function of a  and b , where =−qdqqdnqqdnqqdCACACACRccM1,   =−qnqqnnqqnnqqnCACACACSccM1 The following result is crucial to the development of our design method. Lemma 2. If there exists a symmetric positive definite matrix P, such that the following matrix inequalites are satisfied  01),(),(<−−−−baCPABbaCPBAPPAAqqTqTqqTqqTq      (13)                                                                      3 01 ≥−qTqPBB            (14) then all zeros Q(z) (11) are inside the unit circle.  Theorem 1. If there exist coefficients a and b and a symmetric positive definite matrix P such that the LMIs (13) and (14) are satisfied, then the controller (2) is a low-order stabilizing controller for system (1).     Theorem 1 gives a practical method for  finding a low-order stabilizing controller. The LMIs (13) and (14) together with a set of linear constraints on Cq(a,b) can be solved as a feasibility problem using a convex programming toolbox such as [2]. For each desired order o f the controller, the feasibility problem can be solved to see whether such a low-order control exists. Thus this algorithm for computing the coe f ficients a and b to make Q(z) outer function is convex, versus a non-convex problem without using coprime factors.    Theorem 1 can be used as a building block for solving more complex design problems. We will develop the solutions to a few o f these problems in the following sections.  3.  Pole-Placement Design With a low-order controller, we no longer have the f reedom to arbitrarily place all the closed-loop system poles. The objective then is to find a low-order controller such that the closed-loop system poles. The objective then is to find a low-order controller such that the closed-loop system poles are close to a set of  pre-speci fied poles. Thus the first step is to set the poles of Q(z) (that is, the roots of df(z)dc(z)=0 ) at the desired locations. Then we propose an optimization problem  ∞− ))(1)((min)(zQzWzK             (15) subject to Q(z) being outer function, where W(z) is a stable weighting function, allowing the emphasis of the placement o f the dominant closed-loop poles. We can regard (15) as a regional pole-placement problem, in which the pole-placement regions are determined by the poles of Q(z). We remark that if we set nc=n-1, then (15) becomes the reduced-order observer  design in which all the desired poles can be exactly placed.    Let the minimal realization of the state -space model of W(z)(1-Q(z)) be given by ↔−0),())(1)((baCBAzQzWwqwqwq     (16) where Cwq(a,b) is a linear function of a and b. Using the Bounded- Real Lemma [5], the optimization problem (16) can be expressed as γ)(minzK                   (17) subject to the LMIs 00),(0),(<−−−γγbaCXBBXABbaCXBAXXAAwqwqTwqwqTwqTwqwqTwqwqTwq (18) 0>X                      (19) and the LMIs (13) and (14). This problem is convex and can be solved using an LMI algorithm in the unknowns P, X, a, b, and γ.  4.  Model-Matching Design     In addition to pole-placement design, we can use the low-order controller design idea in Theorem 1 as a building block to obtain sub-optimal designs of many optimal control  problems. One o f the optimal control problems of interest is the model-matching problem, that is, the design of u=-K(z)y for system (1) so that the closed-loop transfer function T(z) (10) matches as closely possible, in the frequency domain, to a desired stable transfe r function Td(z), which is usually a low-order trans fer  function incorporating the desire features o f the control speci fications. Using the coprime factor  for the plant G(z) and the controller K(z), We can define this problem as  ∞∞+−=−))()()()()()()()((min))()()((min)()(zNzNzDzDzNzNzTzWzTzTdzWcgcgcgzKzK                                 (20) where W(z) is a stable weighting function.   The optimization problem (20) cannot be directly set up as a convex optimization problem, because the controller coe f ficients a and b are contained in the denominator Q(z). One way to circumvent this dif ficulty is to formulate a suboptimal control problem by removing the denominator Q(z) and using it for a pole placement design. One such possibility is to pose the suboptimal control problem of ∞+−−)))()()()((1)(())()()()((min)( zNzNzDzDzWzNzNzTzWcgcgcgdzK (21) I f the design (21) yields Dc(z) and Nc(z) such that Q(z)=1, then the controller also satisfies (20). Besides (21), there are many other alternatives to                                                                      4 develop a suboptimal control problem. We will only deal with (21) in this paper. In the model-matching design problem, the poles of Td(z) are the desired dominant poles of the closed-loop system. Thus in performing the coprime factorization o f the plant G(z), )(zdf should contain these desired poles. Construct the minimum state-space realization ↔+−−0),()))()()()((1)(())()()()((baCBAzNzNzDzDzWzNzNzTzWmmmcgcgcgd   (22) where Cm(a,b) is a linear function of a and b.   The suboptimal control problem (21) can be readily solved as an LMI problem of  γ)(minzK                       (23) subject to the LMIs 00),(0),(<−−−γγbaCXBBXABbaCXBAXXAAmmTmmTmTmmTmmTm  (24) 0>X                   (25) and the LMIs (13) and (14) for Q(z). The LMI variables for this problem are γ, a, b, and the positive matrices X and P.  5. ∞H Suboptimal Design Consider the two-input, two-output linear time-invariant system G(z) with the state-space realization )()()()1( 21 kuBkwBkAxkx ++=+       (26a) )()()( 121 kuDkxCk +=η            (26b) )()()( 212 kwDkxCky +=          (26c) where x is the n-dimensional state variable vector, w is the disturbance input variable, u is the control input variable, η is the controlled output variable, and y is the measured output variable. Here the triple (A,B2,C2) is stabilizable and detectable. For system (26), we propose to develop a design algorithm to obtain a suboptimal low-order controller u=-K(z)y to minimize the H∞-norm of the closed-loop transfer function from w to η. To use the result of Theorem 1, we first  develop the appropriate coprime factors. We write (26) in transfer function form =uzDzNzDzNzDzNzDzNyggggωη)()()()()()()()(22211211         (27) where )()()(zdznzNfijnij =,  i , 1=j , 2      (28) and )()()(zdzdzDfgng =                   (29) are stable trans fer functions with dgn(z)=det(zI-A). In this notation, the closed-loop trans fe r function o f (26) f rom w to η, denoted as Tηw(z) ,using the controller coprime factors (5) is  )()()()()()()()(2211zNNzDzDzDzNzDzNzTccgcc++=ηω   (30) where )()()()()()()()( 21122211zdznzDzNzNzNzNzNfng=−=  (31) with )(znn being a polynomial. For the same reason as the model-matching problem, we cannot directly minimize Tηw∞ because with the controller parameters in Dg(z)Dc(z)+N22(z)Nc(z), the resulting problem would be non-convex. Thus we formulate a suboptimal H∞ control problem of  ∞+−+)))()()()((1)(())()()()()((min2211)( zNzNzDzDzWzDzNzDzNzWccgcczK (32) where W(z) is a stable weighting function. Construct the minimum state-space realization  ↔+−+0),()))()()()((1)(())()()()()((2211baCBAzNzNzDzDzWzDzNzDzNzWhhhccgcc  (33) where Ch(a,b) is a linear function of a and b. The suboptimal H∞ optimization problem (32) can be readily solved as an LMI problem of  γ)(minzK                      (34) subject to the LMIs 00),(0),(<−−−γγbaCXBBXABbaCXBAXXAAnhThhThThhThhTh  (35)                                                                      5 0>X                       (36) and the LMIs (13) and (14) for  Dg(z)Dc(z)+Ng(z)Nc(z). The LMI variables for  this problem are γ,  a,  b, and the positive matrices X and P.  6.  Conclusion In this paper we have presented a new low-order controller design method for discrete single-input single-output systems. This method requires only the solution of a convex optimization problem. We established a suf ficient condition which guarantees a trans fer  function being outer. Using the coprime factors and the properties o f discrete outer functions, we show that the design of low-order controller can be formulated as a zero-placement problem. And, the solution can be obtained using LMI algorithms.   References [1]T. Iwasaki and R. E. Skelton, “All Controllers for the General H∞ Control Problem: LMI Existence Conditions and State Space Formulas,” Automatica, Vol. 30, pp. 1307-1317, 1994.  [2] P. Gahinet, A. Nemirovskii, A. J. Laub, and M. Chilali, LMI Control Toolbox, The MathWorks, Inc., 1995.  [3] L.El Ghaoui and V. Balkrishnan, “Synthesisof Fixed-Structure Controllers via Numerical Optimization,” Proc. 1994 CDC, pp. 2678-2683.  [4] K. M. Grigoriadis and R. E. Skelton, “Fixed-Order Controller Problems Using Alternating Projection Methods,” Proc. 1994 CDC, pp. 2003-2008.  [5] P. Gahinet and P. Apkarian, “A Linear Matrix Inequality Approach to H∞ Control,” Int. J. Robust and Nonlinear Control, vol. 4, pp. 421-448, 1994.  [6] J. K. Shiau and J. H. Chow,”Robust Decentralized State Feedback Controller  Design Using an Iterative Linear Matrix Inequality Algorithm,” Preprints of 13th IFAC World Congress, Vol. H, pp. 203-208, 1996.  [ 7] S. Wang and J. H. Chow “Coprime Factors, Strictly Positive Real Functions, LMI and Low-Order Controller Design For SISO Systems”, Proceedings of the 37th IEEE Conference on Decision   Control, Tampa, Florida,  

Coprime Factors, Linear Matrix Inequalities and Robust Low-Order Controller Design

http://tkuir.lib.tku.edu.tw:8080/dspace/bitstream/987654321/7459/1/882213E032027.pdf

[[alternative]]Coprime Factors, Linear Matrix Inequalities and Robust Low-Order Controller Design

Abstract

Similar works

Full text

Available Versions

Tamkang University Institutional Repository