計畫編號：NSC90-2213-E032-007研究期間：200108~200207研究經費：307,000[[sponsorship]]行政院國家科學委員

蕭照焜

計畫編號：NSC90-2213-E032-007研究期間：200108~200207研究經費：307,000行政院國家科學委員

Tamkang University Institutional Repository

???????????????????? ??????H∞????????? H∞ LOW-ORDER CONTROL DESIGN FOR LINEAR DISCRETE-TIME SYSTEMS  ?????NSC 90-2213-E-032-007 ?????90?8?1??91?7?31? ????????? ???????????? ?????????????? ????????????   Abstract- This report proposes a non-iterative computational algorithm for the design of discrete-time fixed order controller for an H∞ optimization problem. Using the coprime factors and pole placement constraints, the fix-order controller design is reformulated as a convex optimization problem. The solutions are obtained using linear matrix inequality techniques. An aircraft model with 3-inputs and 3-outputs is used to illustrate the design algorithm.  Keywords- coprime factorization, robust control, pole placement, linear matrix inequalities, low-order controller design,    1. Introduction  The H? design techniques are broadly used for robust controller design. However, the order of the resulting controllers are much higher then necessary. In practical control designs, low-order controllers are usually desired for system reliability and ease of implementation. The design of fixed- order controller is still an open control problem.    The difficulty is that the design of low-order controller to optimize certain performance involves a bi-affine matrix inequality(BMI), which is non- convex and cannot be solved using the existing convex programming software. Instead of solving directly the BMI problem, several researchers have shown that low-order controllers can be obtained by solving iteratively LMI subproblems, which are convex. These approaches include alternating projection method [5] rank condition minimization method [6] and successive substitution method [4][10][11]. In [13], a low-order controller design method using coprime factors, strictly positive real function (SPR) and LMIs was developed for continuous -time single-input single-output (SISO) systems. This method is expanded to solve the model- matching problem for continuous-time MIMO systems [14]. For discrete time case, low-order robust controller design algorithms using coprime factors, discrete outer functions and LMIs were developed [9]. This report summarizes the results of the development of the low-order controller design for an H? optimization problem. Using the coprime factors and pole placement constraints, the fix-order controller design is formulated as convex optimization problem subject to several LMI constraints. An aircraft model with 3-inputs and 3-outputs is used to illustrate the design algorithm. The paper is organized as follows. Section 2 discusses the structure of the coprime factorization for the low-order controller. In Section 3 we formulate the low-order stabilizing controller design as an LMI feasibility problem. A pole placement design concept is discussed in section 4. The formulation and solution algorithm for an H? optimization problem are presented in Section 5. The application of the proposed design algorithm for the control of the vertical plane dynamics of an aircraft is included in Section 6.  2. Coprime factorization  Consider a linear time-invariant system G(z) with the state-space realization )()()()()1(kCxkykBukAxkx=+=+               (1) where x∈Rn  is the state variable vector, u∈Rm is the controlled input variable, and y∈Rp  is the measured output variable. Assume that the system (1) is stabilizable and detectable. In the packed matrix notation, G(z) is represented by ↔0)(CBAzG                        (2) Since the system (1) is stabilizable and detectable, we perform a right coprime factorization of G(z) to obtain )()()( 1 zGzGzG DN−=                  (3) The state-space realization of GD(z)  and GN(z)  can be represented as −−↔0)()(CIFBBFAzGzGND             (4) where F is a stabilizing full-state feedback gain such that all the eigenvalues of A-BF are inside the unit circle of the z-planr. In contrast to a full-order stabilizing controller K(z) for G(z), whose coprime factorization  1can be readily defined in terms of A,B,C, and a stabilizing observer gain L, for a reduced-order controller we first need to define its structure before performing coprime factorization. Select the reduced-order controller K(z) with p inputs and m outputs to have the structure ↔knckncknckncN DCBAzK )(           (13)  where Ak n c  and Bk n c  are constant matrices determined from the pre-selected denominators dc i (z) . The unknown coefficients of the numerators ki j (z)  are included in Ck n c  and Dk n c . We note that in these realizations, we can make Ak n o  = Ak n c  T , Dk n o  =Dk n c . But we will have no luck for Bk n o  and Ck n o  in general. We further note that both realizations are required for the design of an H? low-order controller in this paper. =)()()()()()()()()(111111zdzkzdzkzdzkzdzkzKmmpmmpLMOML       (5)   We can define the state-space realization for KD(z)  in the similar way. Since KD(z)  is defined in diagonal form, we will only need one realization in our design algorithm where ki j (z)  and di (z)  are polynomials defined as 0,11,,)( ijniniijniniijij bzbzbzk L++= −−      (6) ↔kdkdkdkdD DCBAzK )(           (14)  0,11,)( ininiinii azazzd L++= −−          (7) mi ,,1 L=  ;     (8) pj ,,1 L=where Ak d  and Bk d  are constant matrices, and Dk d =I. The unknown coefficients of the denominators di (z)  are included in Ck d . ai j , l  and bi j , l  =1,…,ni are unknown coefficients to be solved. We can perform a left coprime factorization of K(z) as  )()()( 1 zKzKzK ND−=                 (9) 3. Low-order stabilizing controller design The coprime factors KN(z)  and KD(z)  are stable transfer function matrices with    Consider the closed-loop regulation system in Figure 1 =)()()()()()()()()(111111zdzkzdzkzdzkzdzkzKcmmpcmmcpcNLMOML      (10) K(z) G(z)r_yFigure1: Closed-loop Regulation System =)()(00)()()(11zdzdzdzdzKcmmcDLMOML      (11)  The closed-loop transfer function from the command r to the output y, denoted as T(z), is )()()]()([)( 1 zKzGzKzGIzT −+=       (15) Using coprime factorization of G(z) and K(z), the closed-loop transfer function T(z) is where dc i (z) , j=1,…,m , are predetermined stable monic polynomials. The order of dc j (z)  and dj (z)  are the same. The order of the controller K(z) is the sum of the degree of d1(z) ,…,  dp(z) . For reduced-order controller, the order of the controller is limited to be smaller then n (the order of the plant (1)). If there are no unstable hidden modes in K(z), then (9) is a left coprime factorization. We can use either observable or controllable canonical-form to realize each transfer function ki j (z )/  dc i (z)  in KN(z)  to obtain the state-space realization of KN(z) . That is, if observable canonical-form is used, KN(z)  can be represented as )()()()( 1 zKzQzGzT NN−=              (16) where Q(z) is defined as )()()()()( zGzKzGzKzQ NNDD +=      (17) Using the state-space realizations (4), (13) and (14), a state-space realization of Q(z) can be written as +−−−↔ICCDFCACBBBFABFBAzQknknckdkncknckdkdkd~00000)(  ↔knoknoknoknoN DCBAzK )(           (12)  =qqqqDCBA                   (18) where Ak n o  and Ck n o  are constant matrices determined from the pre-selected denominators dc i (z) . The unknown coefficients of the numerators ki j (z)  are included in Bk n o  and Dk n o . On the other hand, if controllable canonical-form realization is selected, KN(z)  can be represented as where Aq  is stable. We note that the design parameters appear linearly in Cq . The following results are crucial to the development of the design method proposed in this paper.   2Lemma1 : If there exist a symmetric positive definite matrix P, such that the following matrix inequality is satisfied )()()( zKzGzT NN=                    (23) 0<−−−−ICPABCPBAPPAAqqTqTqqTqqTq       (19)  if Q(z)=I is assumed. In this case, the poles of the closed-loop system are exactly the union of the poles of GN(z)  and the poles of KN(z) , which are predetermined by the designer. Thus we propose the following optimization problem to approach the pole placement design 0≥− qTq PBBI                        (20) then all zeros of Q(z) are inside the unit circle of the z-plane. ∞− IzQsK )(min)(                         (24)  Proof : Suppose the inequality (19) is satified, we have subject to the constraint (19) and (20). We can regard (24) as a regional pole-placement problem, in which the pole-placement regions are determined by the poles of GN(z)KN(z) . We remark that Q(z)=I can be achieved for a full order controller due to Bezout identity [13], in which all the desired poles can be exactly placed. 0))(( <−−+−qqTqTqqTqqTqCPABCPBAPPAA     (21)    Let the state-space model of Q(z)-I be given by The inequality (21)can be written as ↔−0))((qqqCBAIzQ               (25) 0)()()(<−++−−−qqTqTqqTqqTqqqqTqqqCPBBICPABPBAPCBAPCBA Using the Bounded-Real Lemma [1], the optimization problem (24) can be expressed as psKγ)(min                               (26) if ?then 0≥− qTq PBBI subject to the LMIs 000 <−−+ICIXBBXABCXBAXXAApqpqTqqTqTqqTqqTqγγ      (27) 0)()( <−−− PCBAPCBA qqqTqqq        (22) Inequality (22) implies that Aq-BqCq is stable?that is, Q-1(z) is stable. Therefore, all zeros of Q(z) are inside the unit circle of the z-plane. 0>X                                   (28) and the LMIs (19) and (20). This problem is convex and can be solved using an LMI algorithm in the unknowns P,X , Ck n c  , Dk n c , Ck d  and pγ . ?  Theorem 1 : If there exist matrices Bk n c , Dk n c , and Bk d  having the controllable canonical realization structure defined in (13) and (14), such that the LMIs (19) and (20) are satisfied, then u=-K(z)y is a stabilizing controller.  5. Optimization Problem ∞H   Consider the robust optimization problem as shown in Figure 2.  Proof : The proof is established by observing that the LMIs (19) and (20) guarantee the stability of the transfer function Q- 1(z). Which implies that the closed-loop transfer function T(z) in (15) is stable.? K(z) G(z)r ye uW1(z)q1W2(z) W3(z)q2 q3 Figure 2: H? optimization problem    Theorem 1 gives a practical method for finding a low-order stabilizing controller. The LMIs (19) and (20) together with the pre-determined structure of Bk n c , Dk n c , and Bk d  can be solved as a feasibility problem using a convex programming toolbox such as [3]. In this paper, we will concentrate on the design of low-order controller for the H? optimization problems.   4. Pole Placement Design From the system block diagram, we have  =urzPzPzPzPeq)()()()(22211211                (29)   With a low-order controller, we no longer have the freedom 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 are close to a set of pre-specified poles. From (16), we observe that where Tqqqq ][ 321=   3=00)()(111zWzP ,  −=)()()()()()(32112zGzWzWzGzWzP 0<−−−IDBDIRCCRACBRCARRAAzrTzrTzrTzrzrTzrzrzrTzrzrTzrzrγγ    (35)      ,  IzP =)(21 )()(22 zGzP −=  0>R                                   (36) and the LMIs (19), (20), (27), and (28). The LMI variables are pγ , γ , the controller parameters ki j ( z ) , and di (z) , and the positive definite matrices P, X, R.   The H? optimization problem is to find a stabilizing controller u=K(z)e such that the closed- loop system is stable and the H? norm of the closed-loop system from input r to the performance (controlled) output q, denoted as Tq r(z)  satisfies  6. Design Example γ<∞)(zTqr                            (30)    In this section, we will apply the proposed design algorithm to design a fixed order controller for the aircraft model AIRC [7]. The model represents a linearized model of the vertical-plane dynamics of an aircraft, and has three inputs, three outputs, and five states. The inputs are spoiler angle (u1 , mea-sured in tenths of a degree), forward acceleration (u2 , in m/sec2 ), and elevator (u3 , in degrees). The states are altitude relative to some datum (x1 , in m), forward speed (x2 , in m/sec), pitch angle (x3 , in degrees), pitch rate (x4 , in deg/sec), and vertical speed (x5 , in m/sec). The three outputs are just the three states, which are to be controlled.  where γ >0 is a pre-specified constant. The transfer function Tq r(z)  is )()]()()[()()( 11211 zKzGzKIzPzPzTqr−++=   (31) Using coprime factorization of G(z) and K(z), the closed-loop transfer function Tq r(z)  is )()()()()()( 11211 zKzQzGzPzPzT NDqr−+=    (32) Obviously, finding a controller to satisfy (30) is a nonconvex problem and is difficult to solve. However, if Q(z)=I is achieved, then the transfer function Tq r(z)  becomes   The design requirements are to achieve a closed-loop bandwidth of about 10 rad/sec, with reasonably damped responses and minimize sensitivity at zero frequency. In order to do this, we select the following weighting functions in the design. )()()()()( 1211 zKzGzPzPzT NDqr +=        (33) The state-space realization of Tq r(z)  can be written as ↔zrzrzrzrqr DCBAzT )(                     (34) ++++++=2222221)6.0(200)60(000)6.0(200)60(000)6.0(200)60()(sssssssW   (37) =knoknogdpgdppzrACBAAA00000121211 =ssssW250002500025)(3                    (38) =knoknogdppzrBDBBB 1211 [ ]knogdpgdppzr CDCCC 121211=  [ ] The functions W1(s)  and W3(s)  penalize the sensitivity and complementary sensitivity function respectively. No constraint on the control is imposed in the design. knogdppzr DDDD 1211 +=  where ↔1111111111 )(ppppDCBAzP  The full order H? controller has an order of eleven. We aim at designing a third order controller directly while maintaining the closed-loop stability and performance requirements. To illustrate the design algorithm proposed in the paper, we first discretize the system model with a sampling rate of 100 Hz. Following the design procedure discussed in the previous sections. The controller is obtained as ↔gdpgdpgdpgdpD DCBAzGzP1212121212 )()(  Note that we use observable canonical-form realization for KN(z)  here. Now, the design para- ters appear linearly in Bz r  and Dz r . Thus, we formulate the H? optimization problem as ↔kkkkDCBAzK )(  )(min)( psKγγ +  −−−−=6439.00037.00039.00037.05949.00003.00039.00003.05791.0kA  subject to the LMIs  4 −−−−−=3013.00316.05206.11193.00071.00106.26078.00039.06773.1kB  −−−−−=537.150437.061.1832872.97007.07859.0975.4426.664368.0kC  −−−−−−−=73.3388189.012.9628739.6358.15305.45900.187014.04.365kD    The open loop system poles, the selected desired poles and the closed-loop system poles are listed in Table 1. The desired poles are selected from the dominant poles of the closed-loop system with full order controller. The results show that the closed-loop system (with the obtained third order controller) poles are very close to the selected desired poles. The H?-norm bound γ of (30) is 0.97 in this design. Figure3: Singular value plot of the sensitivity function   Open-loop poles desired poles Closed-loop poles1.0 -0.9922+0.0102i -0.9922-0.0102i -0.9998+0.0002i -0.9922-0.0002i 0.7594     0.7603 0.7605 0.7605 0.9144+0.0703i 0.9144-0.0703i 0.9145+0.0704i 0.9145-0.0704i 0.6584     0.7116 0.7199 0.8309 0.926+0.0672i 0.926-0.0672i 0.9218+0.0692i0.9218-0.0692i Figure4: Singular value plot of the complementary sensitivity function Table 1: Pole locations of open loop system, desired poles, and closed-loop system poles.     The singular value plots of the sensitivity function and complementary sensitivity function are shown in Figure 3 and Figure 4 respectively. The results show that the design specifications are satisfied with the third order controller.   Time simulation of the closed-loop system are performed for a unit step change in each of the reference inputs. The system responses are shown in Figure 5. The results show that the controller provide good transient and steady-state performance.  7. Conclusions  Figure5: Closed-loop step response to step demand   This research develops a reliable and systematic low-order controller design method for an H ? optimization problem. Using the coprime factors and pole placement constraints, the fix-order controller design is formulated as convex optimization problem subject to several LMI constraints. The solutions are obtained using LMI techniques. The design algorithm is successfully applied to the control of the vertical plane dynamics of an aircraft.  8. Acknowledgment    The work was supported by National Science Council (Taiwan, Republic of China), under grant NSC-90-2213-E-032-007 .  Reference  [1] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994.  5 6[2] T. Iwasaki and R. E. Skelton, “A Unified Approach to Fixed Order Controller Design via Linear Matrix Inequalities,” Proc. 1994 ACC, pp. 35-39. [3] P. Gahinet, A. Nemirovskii, A. J. Laub, and M. Chilali, LMI Control Toolbox, The MathWorks, Inc., 1995. [4] L. El Ghaoui and V. Balkrishnan, “Synthesis of Fixed-Structure Controllers via Numerical Optimization,” Proc. 1994 CDC, pp. 2678-2683. [5] K. M. Grigoriadis and R. E. Skelton, “Fixed-Order Controller Problems Using Alternating Projection Methods,” Proc. 1994 CDC, pp. 2003-2008. [6] 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. [7] J. M. Maciejowski, Multivariable Feedback Design, ADDISON WESLEY, 1989. [8] K.D.Minto, “A Design method for Low Complexity Control laws,” Proc . of 1st IFAC symposium on Design Methods of Control Systems, ETH ,Zurich ,Switzerland , 1991,pp. 106-111. [9] J.-K. Shiau, J. H. Chow, and T.-L. Chuang “Low-order controller design using coprime factors and linear matrix inequalities for single-input-single-output discrete time systems,” submitted to International Journal of Control, 2002. [10] 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. [11]J.-K. Shiau and J. H. Chow, “Structurally Constrained Suboptimal Control Design Using an Iterative Linear Matrix Inequality Algorithm Based on a Dual Design Formulation”, Tamkang Journal of Science and Engineering, Vol. 1, No. 2, pp. 133-143, 1998 [12]M. Vidyasagar, ControlSystem Synthesis : A Factorization Approach, MIT Press, Cambridge, MA, 1985. [13]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 and Control, Tampa, Florida, Dec. pp.2738-2744 1998. [14]S. Wang and J. H. Chow, “Low-Order Controller Design for Model-Matching Optimization Using Coprime Factors and Linear Matrix Inequalities,” Proceeding of American Control Conference, 1999,Vol.3,pp.1871-1875,1999. 

H-Infinity Low-Order Controller Design for Linear Discrete Systems

http://tkuir.lib.tku.edu.tw:8080/dspace/bitstream/987654321/7485/1/902213E032007.pdf

[[alternative]]H-Infinity Low-Order Controller Design for Linear Discrete Systems

Abstract

Similar works

Full text

Available Versions

Tamkang University Institutional Repository