A Simple Approach to Synthesizing Naïve Quantized Control for Reference Tracking

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1 A Simple Approach to Syntheizing Naïve Quantized Control for Reference Tracking SHIANG-HUA YU Department of Electrical Engineering National Sun Yat-Sen Univerity 70 Lien-Hai Road, Kaohiung 804 TAIAN Abtract: - Thi paper preent a ytematic approach to deigning a naïve quantized control law that combine a regular round-to-nearet quantizer with a cloed-loop dynamic ytem to generate a coarely quantized control force for reference tracking. The naïve quantized control law i derived from minimizing an intantaneou abolute value of a performance-index ignal. The tability and performance of the derived quantized control law are analyzed, and it relation to internal model control i clarified. Key-ord: - Quantized control, internal model control, bang-off-bang control. Introduction The deign of a quantized control law involve the determination of optimal quantization level for the control force to perform a certain control tak. Variou method have been developed to derive quantized control law, for example, quantized control derived by dynamic programming [,], alh function [3], LQG optimization [4,5] and receding horizon control [,7]. However, thee quantized control law uffer from one or more of the following problem: () Stability i not proved; () The control law require heavy computation. The implet type of quantized control i a naïve quantized control law, which generate the control force at every intant by directly rounding a linear feedback control to the nearet level in the admiible et. Thi paper preent a practical naïve quantized control law, upplying rigorou analyi for it tability and performance. The circuit realization of the propoed quantized control law require only filter in combination with a imple round-to-nearet quantizer. Quantized Control Synthei a a Signal Minimization Problem Conider the deign of a quantized control law for reference tracking. Given a pecified et S of quantization level, the control ignal u i elected from S at every intant of time o that the plant output y follow the reference ignal r a cloely a poible. In thi paper, thi goal i achieved by minimizing a pecified performance index. A a Fig. : Related Signal Minimization Problem: Quantized control u i yntheized to make error ignal e a mall a poible. performance index, an error ignal e to be minimized i choen to be a frequency-weighted um of the tracking error y-r and control force u. hopefully, a compromie i achieved between minimizing y-r for good performance and minimizing u for tability and efficiency. An optimal control law can thu be derived by olving a related minimization problem hown in Fig., filter and put different frequency-dependent weight on y-r and u, repectively. The value of u at each intant of time i optimally choen from et S to minimize the intantaneou abolute value of e. 3 Optimal Solution: Naïve Quantized Control Suppoe that plant P and weighting function and have the following minimal repreentation. ISBN: ISSN:

2 Fig. : Naïve Quantized Control Sytem. x& = Ax+ Bu P : y= Cx x& = A x : e = Cx x : & e + B + D = A x = C x ( y r) ( y r) + B u + D u () () (3) ithout lo of generality, D = i aumed (thi i alway poible by caling and in the ame proportion to make D =). The error ignal e can thu be written a, e = e + e = u z (4) z= C ( C x (5) x+ D r y) The optimal olution to minimizing e i, u= Q(z) () Q(z) perform direct quantization, quantizing z to the nearet level in et S. The optimal control law i a naïve quantized control with output feedback, a diplayed in Fig.. 4 Stability and Performance Thi ection give a detailed analyi of the tability and performance of the naïve quantized control ytem. 4. Internal Stability e are concerned with the tability of the ytem in the preence of tiny circuit noie. Fig. 3: Linear Sytem Analyi: The quantizer i modeled a a ource of additive noie when the quantizer input z i within the quantization range. Definition. A naïve quantized control ytem i aid to be internally table if all it internal ignal are bounded under the influence of infiniteimal noie of arbitrary pattern added every in the ytem. Theorem (Stability condition) ith ymmetric uniform quantization S={ M, M +,, M }, the naïve quantized control ytem in Fig. i internally table if (C) The quantizer input z doe not exceed the quantization range, i.e., z ( M ). (C) The zero of + P have negative real part and no pole-zero cancellation occur in Re 0 in forming P Proof. From (4), the quantizer in Fig. can be replaced by a ource of additive noie a hown in Fig. 3. If the quantizer input z doe not exceed the quantization range, then the quantization noie will be bounded by half of the quantization tep, i.e., e 0.5. In thi cae, the naïve quantized control ytem can be thought of a a linear feedback ytem ubject to noie e and other infiniteimal exogenou noie. Therefore, according to linear feedback theory [8], Condition (C) guarantee the boundedne of internal ignal. 4. Signal Tracking and Noie Attenuation The output can be derived in relation to input r and quantization error e. From Fig. 3, P P Y = R+ E (7) + P + P Y, R and E are the Laplace Tranform of y, r and e, repectively. The firt term on the right-hand ide of Eq. (9) i related to input reference r, the input-output tranfer function hould be a cloe to unity a poible over the input bandwidth. The econd term on the right-hand ide of Eq. (9) i a noie term; the tranfer function from quantization ISBN: ISSN:

3 Fig. 4: Equivalent Internal Model Control Structure noie e to r hould be a mall a poible to eliminate noie. To thi aim, we may deign to approximately invere P and make very large within control bandwidth. 5 Internal Model Control Interpretation The tabilization mechanim of naïve quantized control can be further explained uing the internal model control concept [9]. The quantizer input z in Fig. 3 can be expreed in -domain a. H ( ) =, P+ Z = HR GE (8) G( ) = (9) P+ According to (4), () and (8), the naïve quantized control ytem can be redrawn a the internal model control tructure hown in Fig. 4. In the diagram, a pure unity gain which i regarded a a model of the quantizer i ued to cancel the effect of the quantizer. The quantization error e, the magnitude of which reflect the amount of the model error, i bounded by half the maximum quantization tep for arbitrary z within the quantization range. Therefore, in the quantization range, the ytem i effectively open-loop, quantization error e i regarded a a bounded diturbance and the linear part of the ytem G and H determine the tability. The requirement for the quantizer input not exceeding the quantization range and the linear part of ytem G and H being internally table turn out to be the tability condition tated in Theorem. Note that, the departure of the quantizer from the pure unity-gain model grow with the increae of the quantizer input level beyond the quantization range. In thi cae, the tability of the naïve quantized control may not be guaranteed. Thi property impoe a limitation on the allowable input range of the native quantized control ytem. Corollary (Allowable Input Range) Aume that all initial condition are zero. ith ymmetric uniform quantization S={ M, M +,, M }, the naïve quantized control ytem in Fig. atifying the condition (C) in Theorem i table if the reference input r atifie the level contraint, r( t) H M + ( G ), t (0) tranfer function H and G are hown in (9), and H denote the abolute integral of the impule repone of the tranfer function H. Proof. According to the linear ytem theory [8], (8) implie the following inequality, z H r + G e, t () r repreent the peak abolute value of ignal r. Making the right-hand ide of () le than or equal to M and ubtituting 0.5 for e yield the contraint (0) which prevent quantizer overload. Choice of eighting Function i deigned mainly for performance. For good reference tracking and noie attenuation, hould be large within control bandwidth. One imple deign i to chooe a the invere of an elliptic filter, with the filter topband et equal to the precribed control bandwidth. In thi deign, will have pole optimally pread over the jω-axi to achieve high gain within the control bandwidth. i deigned to compenate the plant dynamic and enure the tability of the ytem. One imple method for chooing i to et it denominator equal to the denominator of P and elect it numerator to have all zero of + P in the open left half -plane. There are everal problem with the idea of incorporating the plant model in to cancel the plant dynamic. Firt, it i ordinarily difficult to realize with it pole exactly coinciding with the pole of P, owing to the inaccuracy or drift of circuit component and the error of plant model. The incomplete cancellation of the plant dynamic may caue a large tranient ocillation when the plant ha ome lightly damped ISBN: ISSN:

4 Fig. 5: Deigned Naïve Quantized Control with Dwell Time. Table : RMS Percentage of Tracking Error veru Input Frequency Frequency (rad/ec) Tracking Error (%) pole. Second, thi way of cancelling the plant dynamic i not even acceptable for an untable plant, ince it reult in untable pole-zero cancellation. One olution to the problem i to realize a a linear combination of the plant tate variable x. That i, x& = Ax+ Bu : e = C x+ D u. Thi lead to the following naïve quantized control with tate and output feedback. ( z) () u= Q, (3) z ) = C x + D ( r y C x (4) In thi way, it promie an exact pole-zero cancellation in P and avoid intability caued by direct untable pole-zero cancellation. Equation (4) aume that all the tate are available. If the plant tate are not directly meaurable, an oberver can be ued to etimate the plant tate, and the tate etimate intead of the true tate i then ued to contruct ignal z in (4). 7 Deign Example The following example i given to illutrate the deign of the propoed naïve quantized control. Example (Bang-Off-Bang Control). Deign a three-level quantized control law to tabilize the marginally table plant, P = 0, + 0 and to be capable of tracking the reference ignal over the frequency band of 0~00 rad/ec. The quantization level et i S={, 0, }. Deign of. The weighting function i deigned to be the invere of the third-order highpa elliptic filter of the paband ripple db, topband attenuation 50 db, and paband edge frequency 800 rad/ec. Uing the MATLAB function, we obtain, = 40 [A,B]=ellip(3,,50, 800,'high', ''); =tf(40*b,a); Deign of. The weighting function i deigned to have the identical denominator to that of the plant and it numerator i choen to make all zero of + P in the left-half -plane. = To avoid untable pole-zero cancellation, - in the inner loop of Fig. can be realized a, 4 0 = ( P) + 8P Deign of pre-filter F. Obervation find that, when i elected to proper but not trictly proper, tranfer function H will alo be proper but not trictly proper and thu yield a non-zero high-frequency gain. Conequently, a uggeted in (8), an abrupt change of the reference input r will eaily make z wing beyond the quantization range and caue quantizer overload. The poible remedy i to place a lowpa pre-filter in front of the quantized control ytem to filter out the exceive high-frequency content of the input ignal. Here, the pre-filter F i choen a a econd-order lowpa Butterworth filter of cutoff frequency 000 rad/ec. F = Finite Switching Frequency. Becaue of finite peed of the witching device in real application, it a ISBN: ISSN:

5 common practice to realize the quantizer uing clocked or hyterei comparator to et a minimum time duration between two conecutive witching of the quantized control force; thi minimum time duration i commonly referred to a the dwell time in the witched control literature [0]. In thi example, we et the dwell time equal to 0 µec. The reulting naïve quantized control ytem i diplayed in Fig. 5, the dwell time i et by the period of a clock ignal. The quantization only perform at each riing edge of the clock. The quantized control force i held contant in between two conecutive riing edge. Simulation Reult. The naïve quantized control ytem i imulated with inuoidal ignal of different frequencie a the tet input. The amplitude of the tet input are all 0.9. The tracking error i defined a y r in Fig. 5. The performance index i the RMS percentage of the tracking error, which i defined a the ratio of the root-mean-quared value of the tracking error y r and the reference ignal r. The imulation reult are lited in Table. It how that the tracking error for the reference ignal within the control bandwidth are all below one percent of the reference ignal. 8 Concluion A novel naïve quantized control law i derived from minimizing an error ignal which i a frequency-weighted um of the tracking error and control ignal. The tabilization mechanim of the naïve quantized control i nicely explained by connecting it with the famou internal model control. Some of the important feature of the preented naïve quantized control are (a) eae of deign (b) imple circuit realization uing filter and a round-to-nearet quantizer (c) the ability to track a reference ignal and eliminate the quantization noie within the control bandwidth. for multidimenional nonlinear proce by interated dynamic programming, IEEE Tran. Sytem, Man, and Cybernetic, Vol. 3, No., pp. 85-9, 973. [3] C.F. Chen and C.H. Hiao, Deign of piecewie contant gain for optimal control via alh function, IEEE Tranaction on Automatic Control, Vol. 0, No. 5, pp. 59-0, 975. [4] R.E. Laron, Optimum quantization in dynamic ytem, IEEE Tranaction on Automatic Control, Vol., No., pp. -8, 97. [5] T.R. Ficher, Optimal quantized control, IEEE Tranaction on Automatic Control, Vol. 7, No. 4, pp , 98. [] D.E. Quevedo, J.A. De Doná, and G.C. Goodwin, On the dynamic of receding horizon linear quadratic finite alphabet control loop, Proceeding of the 4 t IEEE conference on Deciion and Control, Vol. 3, pp , 00. [7] B. Picao, S. Pancanti, A. Bemporad and A. Bicchi, Receding Horizon control of LTI Sytem with quantized input, Proc. IFAC Conf. on Analyi and Deign of Hybrid Sytem, pp , 003. [8] J.C. Doyle, B.A. Franci and A.R. Tannenbaum, Feedback Control Theory, Macmillan Publihing Company, 99. [9] M. Morari and E. Zafiriou, Robut Proce Control, Prentice-Hall, 989. [0] H. Ihii and B.A. Franci, Stabilizing a linear ytem by witching control with dwell time, IEEE Tranaction on Automatic Control, Vol. 47, No., pp.9-973, 00. ACKNOLEDGEMENT Thi work i upported by the National Science Council, Taiwan, under Grant NSC9- -E Reference: [] R.M. Havira and J.B. Lewi, Computation of quantized control uing differential dynamic programming, IEEE Tranaction on Automatic Control, Vol. AC-7, pp. 9-9, 97. [] J.K. Arora and D.A. Pierre, Optimal trajectorie ISBN: ISSN: