A Method of Setting the Penalization Constants in the Suboptimal Linear Quadratic Tracking Method

Transcription

1 XXVI. ASR '21 Seminar, Instruments and Control, Ostrava, Aril 26-27, 21 Paer 57 A Method of Setting the Penalization Constants in the Subotimal Linear Quadratic Tracking Method PERŮTKA, Karel Ing., Deartment of Control Theory, Institute of Information Technologies, Tomas Bata University in Zlín, nám. T.G. Masaryka 275, Zlín, , Czech Reublic kerutka@ft.utb.cz, htt://.utb.cz Abstract: In this aer, a simle solution of setting the enalization constants in the quadratic functional of the subotimal linear quadratic tracking method is resented. Subotimal linear quadratic method gives very good results in the adative control systems in comarison ith the dynamics inversion method and the ole lacement method and allos us to reduce the order of the controlled system. The enalization constants in this method had to be chosen manually. In this aer, one of the ossibilities, ho to set the constants of the quadratic functional, is given. It is based on a standard regression model. This roosed method has been verified by simulation in Matlab. The simulation results demonstrate the usefulness of the roosed method. Keyords: dynamics inversion method, enalization constants, ole lacement method, subotimal linear quadratic tracking 1 Introduction SISO (single inut, single outut) system is one of the most frequently used systems for control, esecially hen comuters drive the control rocess. It has an otimal behavior and a lot of methods for control have been introduced and verified at the SISO systems (Balátě 1996). When the standard controllers ith fixed, though adjustable, arameters are unsatisfactory, a self-tuning controller (STC) is an alternative for control (Bobál et al. 1999, Bobál et al. 2). The main idea on hich an STC is founded is a combination of a recursive identification rocedure and a choice of controller synthesis. The advantages of SISO system together ith the adative controller can be alied to the decentralized control. Perůtka (2) comared three techniques for the controller synthesis, the dynamics inversion method (Bobál et al. 2), the ole lacement method (Bobál et al. 1999) and the subotimal linear quadratic (LQ) tracking method (Dostál and Bobál 1999). All of them give good behavior, but only the subotimal LQ tracking method allos us good control of reduced order system (Perůtka 2).This method as introduced by Dostál and Bobál (1999). A comutation of all oles in every counting eriod and stabilizing controller for unsteady systems are the big advantages of this method. Hoever, the enalization constants in the quadratic functional have to be set manually. In this aer, one of the ossibilities, ho to adjust these constants is given. This ne method is based on a standard regression model and reflect the influence of overshooting of the outut signal. Firstly, the subotimal LQ tracking method is briefly given. This is folloed by the descrition of the roosed method. Finally, simulation result of the roosed method is shon

2 2 Subotimal LQ tracking method LQ tracking means the linear quadratic deterministic control and e reduce the roblem into the subotimal tracking. There are to ays of solving this roblem (Dostál and Bobál 1999). The first one is based on an estimation of state-sace vectors and has to solve the Riccati equations. Another ay uses the inut-outut descrition, the external linear model (ELM) of the controlled system. Sectral factorization and other comlex oerations ith the multinominals are alied in this method. It is requested a feedback controller Q as the controlled system ould be stable in the otimal tracking roblem. Thus, e ant to minimize the quadratic functional J 2 { () + 2 µ e t ϕu () t } dt (1) here µ, ϕ > are the enalization constants. This functional can be ritten in agreement ith the Parseval theorem as j * { e () s µ e() s + u () s ϕu() s } 1 * J 2πj j ds (2) The controller synthesis ith the otimal LQ tracking method gives a controller, hich is ractically nonroductive because of the deendence of the arameters on the initial values. Therefore, the controller synthesis method giving the subotimal tracking of the reference signal for a grou of reference signals ill be derived. Let us suose the Lalace form of the reference signal funtion as here h () s sf ( h ) deg( ) f () s () s deg, () f (3) (4) Figure 1 - Block scheme of the 1 DOF control Comensator Q(s) is an integrator in Figure 1. Lalace form of the signals in the controlled system is b o y y u +, u fu ou a a, q u e (5) here o y are initial conditions for y and o u are initial conditions for u. We otain equations - 2 -

3 e u d q d ( af ) bo u fo y ( af ) bo u fo y (6) (7) After subtituting s instead f and (3) into (6) and (7) e obtain e m f d (8) here u q m f d as + bq d m ah bf o sf o u y (9) (1) (11) We obtain a conroller via subotimal tracking method, as is folloed. We ill count stable multinominals g and n as a result of the setral factorizations * * * ( as) ϕas + b µ b g g n * n a * a (12) (13) Multinominals q, in the controller transfer function are given ith solving these diohantic equations g * q - v * as b * µn (14) g * v * b (as) * ϕn (15) here deg (v) < deg (g), suosing that f solving only the one olynomial equation does not divide a, e can reduce the roblem on as + bq gn (16) 3 A method of setting the enalization constants ϕ, µ of the functional J for the subotimal LQ tracking method Let us suose f(ϕ,µ) (17) here is a maximum value of the overshooting of the outut signal in the ste resonse of the hole system, ϕ, µ are the enalization constants of the functional J. Furthemore, let us suose the global minimun of the equation (17). This function deendence for the system given in the simulation results section is shon in the table and figures belo

4 Table 1 - Deendence of the overshooting on the enalization constants µ µ,4,2 4,2,5,3 5,3,6,4 6,4,7,6 7,6,8,7 8,7 1,9 1,9 2,17 2,17 3,22 3,22 4,23 4,23 1,34 1,34 1,47 1,47 1,61 1,61 ϕ 1 ϕ 1,4,3,2,1,12Ln(µ) +,99 r, µ Figure 2 - Deendence of the overshooting on the enalization constant µ for ϕ 1,4,3,2,1,12Ln(µ) -,137 r, µ Figure 3 - Deendence of the overshooting on the enalization constant µ for ϕ 1-4 -

5 As e can see from Table 1 and Figure 2 and 3, e can choose one enalization constant, e.g. ϕ 1 and the second one change. Intersection of the logarithm regression curve ith the x axis, hich is used for roortional systems, gives us a first estimation of the otimal constant. If the overshooting still remains e ill do the correction ith the equation () k,8µ ( k 1) µ, k 1,2 K here k is the correction ste. (18) 4 Simulation examle Simulation as realized on the roortional system ith transfer function G () s s 2 6,2 + 12,2s ,7364 (,11s + 1)(,4544s + 1) (t), y(t),.4 y(t) (t) time (s) Figure 4 - Subotimal LQ tracking ith the enalization constants set manually (t), y(t),.4 y(t) (t) time (s) Figure 5 - Subotimal LQ tracking ith the enalization constants set by the roosed method - 5 -

6 .4 1 (t), 2 (t) y 2 (t) y 1 (t) čas (s) Figure 6 - A comarison of the both methods 5 Discussion In this aer, a method for setting the enalization constants of the quadratic functional of the subotimal LQ tracking method is shon. It is clear from Table 1 and Figure 2 and 3, that is suficient to count only one enalization constant, hile the other one can be chosen, the regression equations in Figure 2 and 3 are the same. The main advantage of the roosed method is, that this method seed u the control (see Figure 6). Red curve in Figure 6, hich is the detail of the Figure 2 and 3, reresents the outut signal ith the roosed algorithm and the green one the standard subotimal LQ tracking algorithm. 6 Conclusions The subotimal LQ tracking method as introduced by Dostál and Bobál (1999) and gives very good behavior. The method roosed in this aer seed u the control of this method. Sectral factotization is an imortant asect for stabilization of the system. Another advantage is its accurancy hich as verified by simulation in Matlab Simulink on a SISO system. These characteristic demostrate the usefulness of the roosed method. References BALÁTĚ, J Selected arts of automatic control. Brno: VUT Brno, 1996 (in Czech). BOBÁL, V., J. BÖHM, J. PROKOP and J. FESSL Practical Asects of Self-Tuning Controllers: Algorithms and Imlementation. VUTIUM Press, Brno: University of Technology, 1999 (in Czech). BOBÁL, V., DOSTÁL, P., MACHÁČEK, J., VÍTEČKOVÁ, M. 2. Self-Tuning PID Controllers Based on Dynamics Inversion Method. In: Proc. IFAC Worksho on Digital Control, University of Catalonia, Sain, 2 DOSTÁL, P., BOBÁL, V The subotimal tracking roblem in linear systems. In: 7 th Conference on Control and Automation, Haifa, Israel, 1999, PERŮTKA, K. 2. The Decentralized adative continuous-time control. Diloma ork. Zlín: VUT Brno, FT in Zlín, 2 (in Czech)