Application of LMI for design of digital control systems

Transcription

1 BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vol. 56, No. 4, 28 Application of LMI for design of digital control systems A. KRÓLIKOWSKI and D. HORLA Institute of Control and Information Engineering, Poznań University of Technology, 3a Piotrowo St., Poznań, Poland Abstract. The paper considers a digital design of time-invariant systems in the case of step-invariant ZOH), bilinear Tustin s) and fractional order hold FROH) discretization methods. The design problem is formulated as linear matrix inequalities LMI). A closed-loop stability of the digitally designed control systems is discussed. Key words: digital control, LMI.. Introduction The equivalence between continuous-time and digital systems takes place when the responses of both systems are closely matched for the same inputs and initial conditions. The digital design approach presented in this paper called digital redesign consists in design a suitable analogue controller first and then convert the obtained analogue controller to the equivalent digital controller maintaining the properties of the original analogously controlled system. Obviously, the stability of redesigned system is required for the practical application of of the digital redesign technique. The objective of this paper is to present and compare the digital redesign methods when the step-invariant, bilinear and fractional order hold transformations are used. The first two methods are most commonly used methods of discretization while the third one is used to obtain the model with zeros as stable as possible. The design method is based on the linear matrix inequality LMI) technique according to the approach presented in. When the optimisation problem is convex, it can be solved effectively and fast with the use of LMI algorithms 2, 3. It can be said that once the problem is formulated as a set of LMIs, it can be treated as solved, and used software such as LMI Toolbox, Yalmip, etc) enables one to find the exact optimum. The result of this paper is the enlargement of the redesign method presented in to the case of bilinear and fractional order hold discretization methods. Performance of the considered design method applied with considered transformations is illustrated by simulation study of fourth-order system and linearized model of chemical reactor for the standard LQR control problem. The asymptotic properties of digital closed-loop systems, i.e. when the sampling period tends to zero are also discussed. 2. Control problem formulation Consider a linear time-invariant continuous-time system described by ẋ c t) = Ax c t) + Bu c t), x c ) = x, ) y c t) = Cx c t), 2) where x c t) is the n-dimensional state vector, u c t) is the m- dimensional control vector, y c t) is the p-dimensional output vector, and A, B, C are matrices of appropriate dimensions. The continuous-time controller is given by u c t) = K c x c t) + E c r, 3) where K c is the state feedback gain matrix, E c is the feedforward gain matrix and r is the p-dimensional reference vector. The system ) and the controller 3) yields the continuous-time closed-loop control system ẋ c t) = A BK c )x c t) + BE c r, x c ) = x, 4) where the output remains as in 2). 2.. Step-invariant transformation. The discrete-time model of the closed-loop system 3), 4) obtained with stepinvariant ZOH) method is given by x c kt + T) = G c x c kt) + H c E c r, 5) where T is the sampling period, and y c kt) = Cx c kt), 6) G c = expa BK c )T), 7) H c = G c I)A BK c ) B. 8) It is to be stressed that, whenever identity matrix is used, its size results from matrix algebra. The corresponding matrix transfer function is as where K si z) = CzI G c ) H c E c. 9) The state Eq. ) with a digital control input can be given ẋ d t) = Ax d t) + Bu d t), x d ) = x, ) u d t) = u d kt) = K d x d kt)+e d r, y d t) = Cx d t), ) kt t < kt +T, 2) Andrzej.Krolikowski@put.poznan.pl 36

2 A. Królikowski and D. Horla where K d and E d denote the digital feedback and feedforward gains, respectively. The resulting closed-loop system is ẋ d t) = Ax d t) BK d x d kt) + BE d r, 3) and the discrete-time model of the closed-loop system 3) at t = kt + T is given by where x d kt + T) = G HK d )x d kt) + HE d r, 4) y d kt) = Cx d kt), 5) G = expat), 6) H = G I)A B. 7) If A is singular, the matrix H can be computed as H = i! AT)i BT. i= Now the problem is formulated as follows: for a well designed gains K c, E c in continuous-time controller 3), determine the digital control gains K d, E d in ) such that digitally controlled system 2) is stable and the output of discrete-time system 4), 5) matches the output of continuous-time system 4), 2) as closely as possible Bilinear transformation. The discrete-time model of the closed-loop system 4) obtained with bilinear transformation Tustin s method) is given analogously to 5), 6) by x c kt + T) = G c x c kt) + H c E c r, 8) y c kt) = C c x c kt) + D c E c r, 9) where 2 G c = I T ) 2 A BK c) I + T ) 2 A BK c), 2) H c = T I T c)) 2 2 A BK B, 2) C c = 2C I T c)) 2 A BK, 22) D c = T 2 C I T 2 A BK c)) B. 23) The matrix transfer function for 8), 9) is K bl z) = C c zi G c ) H c + D c E c. 24) The matrices G and H in the corresponding discrete-time model of the closed-loop system 4) are now given by G = I T2 ) I A + T2 ) A, 25) H = T 2 I T 2 A ) B. 26) The following asymptotic property of step-invariant and bilinear transformations has been proved in 4: suppose that in the stable closed-loop system 4) the matrix A BK c is diagonalizable with all real eigenvalues, then lim T + K si K bl lp = 27) for all p, where lp denotes the l p -induced norm Fractional order hold transformation. The discretetime model of the closed-loop system 4), 2) obtained with fractional order hold FROH) method is given by 5 x c kt + T) x kt + T) where G c is given by 7) and γ = γ = G c βγ = γ βγ + E c r, x c kt) x kt) 28) y c kt) = Cx c kt), 29) expa BK c )s)dsb = H c, 3) s T expa BK c)s)dsb = A BK c ) G c T G c I)A BK c ) B. 3) It can be noticed that γ = H c where H c is given by 8). The parameter β is the device adjustable gain, and in the cases of ZOH and FOH first-order hold) considered as particular cases of FROH, we have β =, β =, respectively. The corresponding matrix transfer function is K fh z) = C zi G c βγ ) γ βγ 32) The corresponding discrete-time model of the closed-loop system 4) is now given by x d kt + T) G γ βγ )K d βγ = x,d kt + T) K d x d kt) γ βγ + E d r, x,d kt) 33) y d kt) = Cx d kt), 34) where G is given by 6) and γ = γ = expas)dsb = G I)A B, 35) s expas)dsb = A T G T G I)A B. From 33) the following equation can be obtained x d kt + T) = G γ βγ )K d x d kt) + γ βγ )E d r + βγ x,d kt), 36) 37). 362 Bull. Pol. Ac.: Tech. 564) 28

3 Application of LMI for design of digital control systems where the corresponding matrix H is now H = γ βγ. 38) It can be noticed that if β = then γ = H where H is given by 7). 3. Design method In this section, the method for determination of the gain matrices K d, E d proposed in will shortly be described. According to the theorem given in if there exist a symmetric positive definite matrix Γ, a matrix F, and a scalar α > such that the following generalized eigenvalue problem GEVP) 2 is solved, then the digital control law 2) with E d = satisfies the given design objective formulated in the problem stated in Section 2.. min α s.t. 39) Γ, F αγ <, 4) G c Γ GΓ + HF αi Γ < 4) GΓ HF Γ where F = K d Γ and denotes the transposed element in the symmetric positions. The feedback gain matrix K d is given then by K d = FΓ. 42) The feedforward gain matrix E d can be obtained by equalizing the steady-state outputs of the original and redesigned systems. For the case of step-invariant transformation, this yields E d = I G HK d )) H) I G c ) H c E c 43) where ) denotes the Moore-Penrose pseudo-inverse, and the matrices G, H are given by 6), 7), respectively. A separate determination of E d is done in order to avoid solving the bilinear matrix inequality BMI) problem. In the case of bilinear transformation, similarly to stepinvariant transformation, the feedforward gain matrix E d can again be obtained by equalizing the steady-state output of the original system y c ) = K bl )r = C c I G c ) H c + D c E c r, 44) and the steady-state output of redesigned sytem y d ) = C c I G HK d )) H + D c E d r, 45) This gives E d = C c I G HK d )) H + D c C c I G c ) H c + D c E c, 46) where G, H are given by 25), 26). In the case of fractional order hold transformation, the steady-state output of the original system y c ) = CI G c ) γe c r, 47) and the steady-state output of redesigned sytem This gives y d ) = CI G + γ βγ )K d γ βγ )E d r + βγ x,d, 48) E d = I G γ K d )) γ I G c ) γe c, 49) so it corresponds to 43). 4. Simulation tests Performance of the described method for both transformations is illustrated through the example of a fourth-order unstable system with the following numerical values : A = 7.5, B =, CT =. The feedback and feedforward gains K c, E c for the standard LQR control problem are K c = 3.623, , , ), 5) E c = 3.623, 5) for the constant reference r =, and where the design parameters in the LQR cost function are Q = I, R =. For all cases β =.5 is assumed. In the first simulation test, the sampling period was taken as T =.2 s, then the digital control gains of the proposed method are obtained K d = , , , ), 52) E d = , 53) K d = 5.974, , , ), 54) E d = , 55) K d = , 3.59, 6.323, ), 56) E d = , 57) for the fractional order hold transformation. Figure shows the errors in between original y c kt) and redesigned y d kt) system step responses for step-invariant, bilinear and fractional order hold transformations and T =.2 s plotted in discrete time instants. Bull. Pol. Ac.: Tech. 564)

4 A. Królikowski and D. Horla Fig.. Errors in step responses between original and redesigned control systems for T =.2 s In the second simulation test, the sampling period was taken as T =.2 s, then the digital control gains of the proposed method are obtained Kd =.6595,.828,.263, 2.399), 58) Ed =.6595, 59) Kd = 4.747, , , ), 6) Ed = 2.996, 6) Fig. 3. Plot of matching error δ versus sampling period T In the second example, the chemical reactor for waste management was taken for consideration. The model was linearized around the equilibrium point yielding the following matrices A = , B =.52, C T =. Kd = 2.564,.975,.9938, ), 62) The feedback and feedforward gains Kc, Ec for the standard LQR control problem are Ed = 2.564, 63) Kc = 32.6, 5.533, ), 64) Ec = 32.6, 65) for the fractional order hold transformation. Figure 2 shows the corresponding errors in step responses. for the constant reference r =, and where the design parameters in the LQR cost function are 2.8 Q =, R = The weights Q and R are chosen such that the control signal flow rate) that is applied to the actuator stays within the given range. The continuous-time control system has been discretized for T = s. then the digital control gains are obtained by the proposed method Fig. 2. Errors in step responses between original and redesigned control systems for T =.2 s In Fig. 3, the plot of matching error δ versus sampling period T is shown for all transformations, where k=n X δ= yc kt ) yd kt ). N k= It can be observed that for small T the performance of stepinvariant transformation is the best, and for T.7 s the closed-loop system tends to instability for all transformations. 364 Kd = 3.69, 5.827, ), 66) Ed = 3.793, 67) Kd = , 3.362, 7.55), 68) Ed = , 69) Kd = 4.75, 2.33, ), 7) Ed = , 7) for the fractional order hold transformation. Bull. Pol. Ac.: Tech. 564) 28

5 Application of LMI for design of digital control systems Figure 4 shows the errors in step responses between the original and redesigned systems for step-invariant, bilinear and fractional order hold transformations. The plot of matching error δ versus sampling period T is shown in Fig. 5 for all transformations. It is worthy to note that in the large range of sampling period the redesigned closed-loop control systems remain stable. 5. Conclusions This paper presents an approach toward a digital redesign of linear time-invariant systems when step-invariant, bilinear and fractional order hold discretization methods are used. The comparative simulation examples of fourth-order unstable system and third-order stable model of chemical reactor are given for the standard LQR control problem. To see the differences between methods more observable, the matching error between the unit step response of the original system and redesigned digital systems was calculated. In this respect, the performance for all transformations is comparable, however for small values of sampling period the performance of redesigned control system with ZOH discretization is superior to that with Tustin or FROH discretizations. The proposed method ensures the stability of the all redesigned control systems for large range of sampling periods. Fig. 4. Errors in step responses between original and redesigned control systems for T =. s Fig. 5. Plot of matching error δ versus sampling period T REFERENCES W. Chang, J.B. Park, H.J. Lee, and Y.H. Joo, LMI approach to digital redesign of linear time-invariant systems, IEE Proc. Contr. Theory Appl. 4 49), ). 2 S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics, Philadelphia, L. El Ghaoui, and S.-I. Niculescu, Advances in Linear Matrix Inequality Methods in Control, Society for Industrial and Applied Mathematics, Philadelphia, G. Zhang, and T. Chen, Comparing digital implementation via the bilinear and step-invariant transformations, Automatica 4, ). 5 R. Barcena, M. de la Sen, and I. Sagastabeitia, Improving the stability properties of the zeros of sampled systems with fractional order hold, IEE Proc. Contr. Theory Appl. 4 47), ). Bull. Pol. Ac.: Tech. 564)