Optimal PI-lead controller design

Optimal PI-Lead Controller Design Mahdi Riyad Issa Enrique Barbieri Electrical Engineering Department Tulane University New Orleans, LA 70118 Abstract control processes have PID feedback, the design of a practical controller that requires less parameter tuning and guarantees a stable system is needed. The goal of this research is to develop a controller which closely approximates the performance of the PID controller using a design framework that is straightforward t o use and for which software packages are already routinely employed. The framework is that of the infinitetime Linear Quadratic Regulator or LQR. In Section 2 we present the details of the design of the PI-Lead controller using the infinite-time or steady-state LQR. Special attention is given to deriving the conditions that ensure controllability. Section 3 illustrates the design procedure with a thirdorder example and compares the controller performance with a conventionally designed PID. Finally, Section 4 concludes the paper and gives suggestions for further research. The design of an optimal MIMO PI-Lead controller is described for linear, time-invariant systems. We consider the Linear Quadratic Regulator (LQR) design framework as a means of optimally selecting the gains of the controller. Therefore, the controller is designed by solving a steady state algebraic Riccati equation. The performance between the PID and Optimal PILead controllers is illustrated and compared through an example. Simulations and implementations of the controllers are done using the M A T L A B and SIMULINK software packages. 1 Introduction Proportional-Integral-Derivative (PID) controllers are widely used in the process industries. Many approaches have been used to determine the PID controller parameters, including Root Locus, frequency domain analysis, and adaptive techniques. The Ziegler-Nichols design and its various modifications are also widely used. However, these techniques generally require considerable experience from the part of the designer, are usually applicable to single-input single-output (SISO) plants, and take several design iterations to achieve satisfactory performance [1]-[4]. The design method to find the best tuning parameters for a (PID) controller could be quite complex. The complexity of the design increases as the order of the system increases. For a MIMO system, there is in addition interaction between the different control loops which affects each other's performance and makes the design practically impossible. For decoupled systems, direct application of a diagonal PID controller may provide satisfactory performance. Otherwise, an inner controller may be required to provide a decoupled or diagonally dominant open loop response. Since the best known controllers used in industrial 0-8186-7352-4/96 $05.00 0 1996 IEEE 2 Optimal PI-Lead Control Consider a linear, time-invariant system modelled by the set of dynamically uncoupled first-order differential equations where in the ith subsystem E ?I?2"' is the state, '~li E ?I?" is the control, y; E is the output and all matrices are constant and appropriately dimensioned. Note that each subsystem is itself a MIMO system. An output tracking control design problem is defined as follows: consider the system (l),(2) and given a set of constant output set-points yip, i = 1 , 2 , ..., N, design the control laws uj, i = 1,2,..., N , so that asymptotic output error regulation is achieved, that is lim ei(t) = 0; t-00 364 A where e i ( t ) = yi(t) - y:p. A MIMO PI controller was considered in [5] using this formulation. Introduce the augmented state vector Controllability Requirement This formulation assumes the pairs {Ai,&} to be where er6'i = Q i , controllable and the pairs {Ai, to be observable, for i=1, ...,N. The following result gives us the necessary and sufficient conditions for the pairs { A i , &} to be controllable. ea}, (3) where 6 is the ouput of the Lead compensator, 9i = ajyj + i j - ~ i 6 i and ai E %rixrs and pi Theorem For simplicity, assume that the spectrum of Ai does not contain any of the pij and that the pij are distinct (these assumptions can be relaxed and slightly modified but more general rank conditions result). Then, the pairs { A i , B i } , i=112,...,N, are controllable if and only if the pairs {Ai, B i } are controllable and the following rank conditions are satisfied: (4) E %'ixriare diagonal matrices with main diagonals ( a i l ,a i a , . . , aivi), and (Al,Pi, , . .. ,pivi),respectively. We also assume that aji and pij are all positive and nonzero scalars. It follows that zi satisfies the differential equation ii . = A j ~+ i Bit+, where Ai - = [ (5) Ai 0 Ci 0 0 aiCi +CiAi i] ;Bj= -Pj [ ';I(,) and, for A = -pi,, rank [ Air, +pi where Ci Bj We impose a performance measure of the form F = (aiCi Proof: (7) rank(W) = nj where w (8) I &] = ni + 2rs = [ + 2ri (13) AIn, -Ai -cj -aiCi-CiAi XI,, 0 0 71 0 AI,, +pa CiBi for A E a(Ai) = spectrum of A i . We therefore consider A c { a ( A i ) }U (0, -pi,). The following are the three possible cases and the resulting rank conditions: (1) A E a(Ai) and A # 0 , A # -pi,. In this case it is straightforward to show that the rank condition 13 reduces to the requirement that the pair {Ai, Bi} be controllable. (2) A = 0. In this case, equation 13 can be reduced to equation 11 since we have assumed that p;, # 0. (3) A E -Pi. From equation 13, where the unique, positive definite, symmetric matrix Pi solves the algebraic (steady state) Riccati equation (ARE) =0 + CiBi that is, that is, + ATPj - PiBiRF'BTPi + Qi + CiAi)(AIn,- rank[XI,,+2,, - A i $ ( t ) = -Rr'BTPizi Pi& (12) The Popov-Belevitch-Hautus (PBH) test is used to obtain the rank requirements. It states that the pair { A i , &} is controllable if and only if [7] to be minimized subject to the dynamic constraints (5)i. In criterion (7) the matrices Qj and are normally assumed to be symmetric, and positive sernidefinite and positive definite, respectively. This is the standard infinite time linear-quadratic regulator (LQR) problem, which, under certain conditions, hm the linear, constant, full state feedback solution [6] GT(t) = -Kizi = -[Kfl) Kj2) Ky)]zi F ] = ri (9) The controller (8) is stabilizing, so that the error e i ( t ) goes asymptotically to zero as required. Integratting both sides of (8), the required MIMO optimal PI-Lead controller is obtained which, by elementary row and column operations can be reduced to equation 12. 365 3 It can be verified that { A ,B} is controllable. We proceed to select Simulation Results The following example will illustrate the performances of the Optimal PI-Lead and conventional PID controllers on a third-order system. All simulations were done using the MATLAB and SIMULINK software packages. Consider the third-order plant described by (see [8], chapter 9) Q=[ (18) and R=0.001 which, together with A and B,are used to determine the optimal LQR gains. The SIMULINK implementation of the closed-loop system is shown in Fig.(4), and Fig.(3) shows its unitstep response. Fig.(5) and Fig.(6) illustrate that the magnitude of the control needed in the PI-Lead controller is quite reasonable and in fact smaller than that required of the PID controller. K G p ( s )= s ( l ]1 0 0 0 0 0 .0001 0 0 .0001 00 0 0 0 10000 00 0 0 0 0 .0001 .0001 + O.ls)(l + 0.2s) where K=100, so that the ramp-error constant of the system is 100. The design objective is to maintain the ramp-error constant at 100, while achieving good relative stability. The PID-Controller desian 4 The conventional design of a PID controller + KDS + GPID = K p An alternative to the PID controller has been introduced that uses the LQR framework to design an Optimal PI-Lead controller. The advantages of the PI-Lead controller are that it is easy to design using available software packages such as Matlab, and that it is applicable without any additional difficulty to MIMO systems. For example, we have used it to control the joints of a two-link manipulator together with an inner feedback-linearizing loop [9]. The disadvantages are that all the state-variables are assumed to be available, hence, an observer is usually required, and, although the design is reasonably robust to small parameter variations, it is susceptible to external noise. We are currently examining the definition and solution of a second static minimization to further reduce the cost subject to a set of linear constraints as was done in [5]. In essence, the minimum cost is given by KI in terms of its PI and PD components leads to the compensated system G(s) = + 5000(1+ K D I S ) ( K ~ ~KSI ) s2(s 5)(s 10) + + where K p = Kp2 +KDIKI; KD = K ~ i K p z Using the root-locus technique one arrives at a satisfactory design with K p = 0.0735, K D = 0.035, and K I = 0.007. A SIMULINK implementation of the PID-compensated system is shown in Fig.(l) and Fig.(2) shows its unit-step response. The Optimal PI-Lead Design N The open-loop system in state-space form is k !I$+[ Conclusions and Further Research i=l (17) which shows that the cost is a function of the constant output set-points which in some applications may be computed at a higher level in the control scheme. Partitioning Pi as In this example, the values for a and p are chosen to be 10 and 100, respectively. The augmented system matrices are 0 1 0 0 0 0 0 1 0 0 50 -15 0 B= 500; 1 0 0 0 0 0 10 1 0 0 -100 0 the minimum cost J* can be, after adjoining a set of linear constraints, minimized by the method of Lagrange multipliers yielding the optimal set-points. x = [ : 0 -50 y = [l 0 -15 5000 : ] (16) U O]X 1, [ 366 Auto-Scale Graph1 Figure 1: Block diagram of the PID controller Figure 4: Block diagram of the Optimal PI-Lead controller Figure 5: Magnitude of the control,u, needed in the 12- 1- 08- 08 - 04- 02- I . 02 04 08 08 1 12 14 I S 18 2 Figure 3: Unit-step response of Optimal PI-Lead controller Figure 6: Magnitude of the control,u, needed in the optimal PI-Lead controller 367 References [l] Radake F., and Iserman R.: A ParameterAdaptive PID Controller with Stepwise Parameter Optimization. Automatica Vo1.23, 1987, pp.449457. [2] Salem A. K. Al-Assadi and Lamya A. M. AlChalabi: Optimal Gain for Proportional- IntegralDerivative Feedback. IEEE Control Systems Magazine, December 1987, pp.16-19. [3] Wang, L.: New Frequency-Domain Design for PID controllers. IEE Proceedings, Con trol Theory and Application, Vol. 142 July 1995. [4] Zhuang, M. and Atherton, D. P. : PID controller Design for T I T 0 system IEE Proc. Control Theory Appl., Vol 141. No. 2. March 1994. [5] Barbieri Enrique: A Multi-Input/Multi-Output PI Controller for Redundant Robots in the Presence of Flexible Disturbances. Optimal Control Applications and Methods, Vo1.15, 1994, pp.35-48. [SI Kirk Donald E., Optimal Control Theory Prentice Hall, Inc, 1970. [7] Kailath Thomas, Linear System Prentice Hall,Inc, 1980 [8] Kuo Benjamin C., Automatic Control Systems Prentice Hall, Inc, 1991. [9] Issa Mahdi R., Optimal P-Lead and PI-Lead Controller Design, Master Thesis, Elctrical Engineering Department , Tulane University, December 1995. 368 View publication stats