Adaptive Predictive Control with Controllers of Restricted Structure

Transcription

1 Aaptive Preictive Control with Controllers of Restricte Structure Michael J Grimble an Peter Martin Inustrial Control Centre University of Strathclye 5 George Street Glasgow, G1 1QE Scotlan, UK Abstract The application of novel aaptive preictive optimal controllers of low orer, that involve a multi-step cost inex an future set-point knowlege, is consiere The usual preictive controller is of high orer an the aim is to utilise simpler structures, for applications where PID controllers might be employe for example A non-linear system is assume to be represente by multiple linear iscrete-time state-space moels, where n of these moels are linearisations of the unerlying non-linear system at an operating point, etermine off-line One extra moel is ientifie on-line The optimisation is then performe across this range of N f + 1 moels to prouce a single low orer control law One avantage of this approach is that it is very straightforwar to generate a much lower orer preictive controller an thereby simplify implementation Also, with respect to the aaptive nature of the algorithm, the solution is rather cautious Each new upate of the controller involves averaging the cost function across both fixe an currently ientifie moels, proviing robust aaptive control action The metho is applie to a piecewise non-linear system, implemente by switching between several linear systems, an results are given 1 Introuction Preictive optimal control is use extensively in inustry for applications such as large-scale supervisory systems [1 Preictive control epens upon the assumption that future reference or setpoint information is available, which may then be incorporate into the optimal control law to provie improve tracking characteristics an smaller actuator changes The best known preictive control approach is probably Dynamic Matrix Control (DMC), which was introuce for complex multivariable plants with strong interactions an competing constraints [2 DMC has been applie in more than 1 plants worlwie an aims to rive a plant to the lowest operating cost The algorithm inclues a steay state optimiser base on the economics of the process so that set points can be manipulate to optimise the total system The focus of this type of commercial algorithm is at the supervisory levels of the control hierarchy where the orer of the controller is not such a problem If preictive 1

2 control is to become wiely aopte at the regulating level there is a nee for low-orer simple controller structures, an this is the problem aresse The preictive control algorithms base upon multi-step cost-functions an the receing horizon control law, were generalize by Clarke an coworkers in the Generalize Preictive Control (GPC) algorithm [3 Future set-point information has been use in a number of Linear Quaratic (LQ) optimal control problems an summarize in the seminal work of Bitmea et al [4 The use of state-space moels for Generalize Preictive Control (GPC) was propose in [5 an extene in [6 Multi-step cost-functions may also be use in LQ cost-minimization problems The solution of the multi-step Linear Quaratic Gaussian Preictive Control (LQGPC) problem, when future set point information is available, has been consiere in [7, when the plant is represente in polynomial matrix form The solution of the LQGPC cost minimization problem for systems represente in state equation form was given in [8 an [9 There are a number of moel preictive control philosophies which employ state equation moels which are relate to these results, such as in [1 The solution strategy followe is to minimise an H2 or LQG criterion in such a way that the preictive controller is of the esire form an is causal A simple analytic solution cannot be obtaine, as in the case where the controller structure is unconstraine [11 However, a relatively straightforwar irect optimization problem can be establishe which provies the esire solution The aim of this paper is to present a new metho of generating aaptive low-orer preictive optimal controllers that coul be use in non-linear control applications This simplification is to be achieve without losing the benefits of either the multi-step criterion or the future set-point knowlege 2 System Moel The system shown in Fig1 is represente by the linear, time-invariant, iscrete-time statespace system representation given below, where the state vector X(t) = [ x (t) x 1 (t) T is a combination of the states for both reference generator an plant: [ A X(t + 1) = A 1 [ X(t) + B 1 [ D u(t) + D 1 [ ξ (t) ξ 1 (t) (21) X(t + 1) = AX(t) + Bu(t) + Dξ(t) (22) [ z 1 (t) = C 1 [ X(t) + 2 v 1 (t) (23)

3 z 1 (t) = CX(t) + v 1 (t), y h (t) = H 1 x 1 (t), r h (t + p) = H r x r (t) (24) The states x 1 (t) R n, x (t) R p, control input u(t) R, white noise isturbance ξ 1 (t) R, observation z 1 (t) R, white output noise v 1 (t) R, inferre output y h R, riving white noise input ξ (t) R, an inferre reference r h (t) R (t) D r + x (t+1) r z -1 x (t) r H r (t) h y (t) h (t) D 1 A r v (t) 1 u(t) B x 1(t+1) x (t) z -1 1 C 1 + z (t) 1 A 1 -K Figure 1: Plant Moel an Reference Generator To prouce the reference signals at {t+1,t+2,,t+p-1}, the x (t) state is create by elaying x r (t) Hence: x (t) = x r (t) x r1 (t) x r2 (t) x r(p 1) (t) A r H r, A = 1, D = 1 D r (25) To prouce the vector of future reference values from the state vector, the following matrixvector prouct is forme: r h (t + 1) r h (t + 2) r h (t + p 1) r h (t + p) = H r 3 x r (t) x r1 (t) x r(p 2) (t) x r(p 1) (t) (26)

4 with an obvious efinition of terms in (27) R t+1,n = H x (t) (27) Also, the matrices C an D are partitione to give C 11 = [, C 21 = [ C 1, [ [ D D 11 =, an D 12 = These partitions are use later, in the efinition of the D 1 system transfer function matrices Having establishe the plant equations, an estimator is require to preict the inferre output for j steps ahea The estimator is state below: G N = N N = Y h t+1,n = H Nx 1 (t) + G N U t,n + N N W t,n (28) Y h t+1,n = y h (t + 1) y h (t + 2) y h (t + N), H N = H 1 A 1 H 1 A 2 1 H 1 A N 1 H 1 B 1 H 1 A 1 B 1 H 1 B 1, U t,n = H 1 A N 1 1 B 1 H 1 A N 2 1 B 1 H 1 B 1 H 1 D 1 H 1 A 1 D 1 H 1 D 1, W t,n = H 1 A N 1 1 D 1 H 1 A N 2 1 D 1 H 1 D 1 3 Preictive control problem formulation u(t) u(t + 1) u(t + N 1) ξ 1 (t) ξ 1 (t + 1) ξ 1 (t + N 1) For a scalar system with white noise input signals, the preictive control performance inex to be minimise can be efine in the time omain as in [12: J = E J t = { lim T 1 2T T t= T J t } N Q j (r h (t + j) y h (t + j)) 2 + j=1 4 N 1 j= R j u(t + j) 2 (31)

5 where E{} is the unconitional expectation operator an y h an r h are the inferre output an reference signals respectively The error an control weightings, Q j an R j, nee not remain fixe over the sequence of j s By expressing the system escription in the state-space form of Section 2 it is possible, with suitable manipulation, to restate the cost function in frequency omain form [13: J p = E { = 1 2πj lim T z =1 1 2T } T X T (t) Q c X(t) + u T (t) R c u(t) + 2X T (t)ḡcu(t) t= T trace{ Q c Φ XX (z 1 ) + 2ḠcΦ ux (z 1 ) + R c Φ uu (z 1 )} z z (32) where Φ XX, Φ uu, an Φ ux are the power spectrums of the state, control input an the crossspectrum of state an control input respectively To obtain the Q c, R c, an Ḡc matrices it is necessary to first partition G = G N, an R = iag{r, R N 1 } to match the partitioning of U t,n into current an future controls: G = [ G N1 G N2, R = [ R R22 where R 22 = iag{r 1,, R N 1 } Noting that Q = iag{q 1, Q N }, H = [ H H N, Q c = H T Q H, Rc = G T Q G + R, an Gc = H T Q G, the efinitions of Rc an G c can now be expresse in terms of these partititions: [ [ Rc1 R c3 G T R c = = N1 QGN1 + R G T QG N1 N2 R c2 G T QG N2 N1 G T QG N2 N2 + R 22 R T c3 G c = [ G c1 G c2 = [ HT QGN1 H T QGN2 The esire matrices are then efine as: an Ḡc = G c1 G c2 Rc2 1 Rc3 T 31 Polynomial H 2 problem solution (33) (34) Qc = Q c G c2 R c2 G T c2, Rc = R c1 R c3 R 1 c2 RT c3 In orer to prouce the optimal control law for the given system, the spectral factors an Diophantine equations below must first be solve: Spectral Factors D cp D cp = B 1p Q c B1p + Ā 1p R c Ā 1p + B 1pḠcĀ1p + Ā 1pḠ c B 1p (35) D p D p = C pc p + A pr f1 A p (36) 5

6 Diophantine Equations z g 1 D cp Gc 1p + F c 1pĀp = ( B 1p Q c + Ā 1pḠ c )z g 1 (37) z g 1 D cp Hc 1p F c 1p B p = (Ā 1p R c + B 1pḠc)z g 1 (38) z g 2 G f 1pD p + ĀpF f 1p = D 12 C pz g 2 (39) z g 2 H f 1pD p C 21z 1 F f 1p = R f1 A p z g 2 (31) The various polynomial matrices are obtaine from the system transfer functions efine below: Resolvent Matrix: Φ(z 1 ) = (zi A) 1 Plant moels: W (z 1 ) = Φ(z 1 )B, W (z 1 ) = C 21 Φ(z 1 )B Disturbance Moels: W (z 1 ) = Φ(z 1 )D 12, W (z 1 ) = C 21 Φ(z 1 )D 12 Reference Moels: Wr (z 1 ) = Φ(z 1 )D 11, W (z 1 ) = C 11 Φ(z 1 )D 11 Letting Āp = (I z 1 A), the right coprime form of W may be written as: W = Ā 1 p B p = B 1p Ā 1 1p (311) where B p = z 1 B Also, the left-coprime forms for W an W may be written as: W = A 1 p B p, W = A 1 p C p (312) Ultimately, the optimal control problem reuces to minimising where J + = 1 2πj z =1 (T + T + ) z z (313) T + = H c 1p([1 + H c 1 1p G c 1pĀ 1 p (I + G f 1pH f 1 1p A 1 p (I + G f 1pH f 1 1p A 1 p C 2p ) 1 Bp K H c 1 1p G c 1pĀ 1 p z 1 C 2p ) 1 G f 1pH f 1 1p )(A p + B p K) 1 D p (314) This is achieve when T + = The optimal feeback control law, K, is therefore: where K c = H1p c 1 Gc 1p an K f1 = G f 1p Hf 1 1p K = K c [zāp + K f1 C 21 + BK c 1 K f1 (315) 6

7 4 Restricte Structure an Aaptive Control 41 Restricte Structure solution The optimal solution to the preictive optimal control problem simply requires T + to be set to zero In the case of a restricte structure control law, it is necessary that (313) be minimise with respect to the parameters of the given controller structure In the following analysis, it will be assume that K is a moifie-pd controller: K mop D = K p + K (1 z 1 ) 1 τz 1 (41) Therefore, the controller parameters of interest in this case are K p an K Making the appropriate substitutions in (314), as in (315), obtain: T + = Hc 1p ([1 + K cs f BK mop D K c S f K f1 )(A p + B p K mop D ) 1 D p (42) where S f = (zāp + K f1 C 21 ) 1 Rewriting K mop D as a rational function, K = K n = K p(1 τz 1 ) + K (1 z 1 ) K (1 τz 1 ) = K pα + K α 1 α (43) where α an α 1 have the obvious efinitions, T + becomes: T + = K nl n1 K L n2 (44) where L n1 = L 1 /(K n L 3 + K L 4 ) an L n2 = L 2 /(K n L 3 + K L 4 ) (45) L 1 = H1p c [1 + K cs f BD p, L 2 = H1p c K cs f K f1 D p, L 3 = B p, L 4 = A p (46) T + is obviously non-linear in K p an K, renering (313) particularly ifficult to minimise irectly However, an iterative solution is possible if the values of K p an K in the enominator of T + are assume known Defining L 1 = H c 1p[1 + K c S f BD p, L 2 = H c 1pK c S f K f1 D p, L 3 = B p, an L 3 = A p T + = L 1K n L 2 K L 3 K n + L 4 K (47) 7

8 Defining L n1 = L 1 /(L 3 K n + L 4 K ), L n2 = L 2 K /(L 3 K n + L 4 K ), T + K n : becomes linear in T + = L n1k n L n2 (48) As T + is a complex function an the complex conjugate is require in (313), the next step is eviently to split the elements of T + into real an imaginary parts, enote by the superscripts r an i : Splitting K n T + = (L r n1 + jl i n1)(k r n + jk i n) (L r n2 + jl i n2) = L r n1k r n L i n1k i n L r n2 + j(l i n1k r n + L r n1k i n L i n2) (49) an substituting: K r n = K pα r + K α r 1, Ki n = K pα i + K α i 1 (41) T + = K p ((L r n1α r L i n1α i ) + j(l i n1α r + L r n1α i )) + K ((L r n1α r 1 L i n1α i 1) + j(l i n1α r 1 + L r n1α i 1)) (L r n2 + jl i n2) (411) Noting that T + T + = T + 2 = (T +r )2 + (T +i )2, it is obvious that the integran of (313) can be represente by a matrix-vector prouct: where an T + T + = [ T +r T +i F = [ T +r T +i [ (L r n1 α r L i n1α) i (L r n1α1 r L i n1α1) i (L i n1 αr + Lr n1 αi ) (Li n1 αr 1 + Lr n1 αi 1 ) [ L r L = n2 L i n2 = (F x L) T (F x L) (412) [ Kp, x = K (413) (414) The complex integral cost is evaluate for z = 1 Hence, the matrices can be expresse as a function of the real frequency variable, ω: J + = 1 2πj = T 2π z =1 2π/T (T + (z 1 )T + (z 1 )) z z (F (e jωt )x L(e jωt )) T (F (e jωt )x L(e jωt ))ω (415) 8

9 The cost function can be optimise irectly, but a simple algorithm is obtaine if the integral is approximate by a summation with a sufficient number of frequency points, {ω 1,, ω k,, ω N } That is: J + N (F (e jωkt )x L(e jωkt )) T (F (e jωkt )x L(e jωkt )) k=1 = (b Ax) T (b Ax) (416) where A = F (e jω 1T ) F (e jω N T ), b = L(e jω 1T ) L(e jω N T ), x = [ Kp K (417) Assuming that the matrix A T A is non-singular, the least squares optimal solution is: x = (A T A) 1 A T b (418) Of course, as the assumption was mae that the solution x was alreay known in the enominator of T +, this is a case where the metho of successive approximation, as in [14, can be use This involves a transformation T such that x n+1 = T (x n ) Uner appropriate conitions, the sequence {x n } converges to a solution of the original equation Since this optimisation problem is non-linear there may be not be a unique minimum However, the algorithm presente in the next subsection oes always appear to converge to an optimal solution in many inustrial examples 42 Aaptive control The aaptive controller to be escribe is base on the multiple-moel version of a restrictestructure optimal controller This version is so calle ue to the use of a set of mathematical moels to represent a single non-linear or time-varying system at ifferent operating points The aim is to prouce a single controller which will stabilise the entire set of moels The cost function employe is a weighte sum of costs for iniviual system representations Let J + j enote the value of (416) for the jth system moel, an let the probability of this moel being the true representation be enote by p j Also, let the b an A matrices in (416) for the jth system moel be b j an A j respectively Then the multiple-moel cost criterion can be written as: 9

10 J + = = n+1 p j J + j j=1 n+1 p j (b j A j x) T (b j A j x) j=1 = (b Ax) T P (b Ax) (419) where A = A 1, b = b 1, P = iag{p 1,, p n+1 } (42) A n+1 b n+1 The solution to this problem is obviously similar to the single-moel case Assuming that A T P A is non-singular, the least squares optimal solution is: x = (A T A) 1 A T b (421) A controller for a non-linear system can then be prouce by efining the first N f linear moels to represent the non-linear system at ifferent operating points The aaptation is introuce by continually upating moel N f + 1 with recursively ientifie parameters an recalculating the values for x online The following successive approximation algorithm as in Luenberger [14, with a system ientification algorithm incorporate, can be use to compute the restricte structure LQG aaptive controller Algorithm 41 (Aaptive restricte-structure control algorithm) 1 Define N (number of frequency points), ω 1,, ω N, τ an N f (number of fixe moels) 2 Initialise K p = K = 1 (arbitrary choice) 3 Define α (z 1 ), α 1 (z 1 ) (using (43)) 4 Compute C n (z 1 ) = K p α (z 1 ) + K α 1 (z 1 ) 5 Compute C (z 1 ) = α (z 1 ) 6 For j = 1 to N f (a) Solve for the spectral factors D cpj an D pj, an the Diophantine equations for G c 1pj, Hc 1pj, F c 1pj an Gf 1pj, Hf 1pj, F f 1pj (b) Create L 1j, L 2j, L 3j, L 4j, L n1j, an L n2j 1

11 (c) For all chosen frequencies, calculate F j (e jωt ) ), L j (e jωt ) ) F j (e jω 1T ) ) L j (e jω 1T ) ) () Assemble A j = an b j = F j (e jω N T ) ) L j (e jω N T ) ) 7 Estimate current A p, B p, an C p polynomials using a recursive least squares algorithm 8 Repeat steps 6(a) to () for the ientifie polynomials 9 Stack the N f + 1 A an b matrices to form A an b 1 Calculate the restricte-structure controller gains, x = (A T P A) 1 A T P b 11 If the cost is lower than the previous cost, repeat steps 8 to 1 using the new C n Otherwise, use previous controller gains to compute the feeback controller C n (z 1 ) = K p α (z 1 ) + K α 1 (z 1 ) an C (z 1 ) = C n (z 1 )/C (z 1 ) 12 Implement controller in feeback loop an go back to step 7 5 Aaptive Control of Switche Linear Moels A ship roll control problem will now be consiere Ship roll control systems are often use on passenger ferries in orer to maintain a comfortable rie for passengers The ship can be moelle by a secon orer transfer function, where the input is fin angle an the output is ship roll angle [15 The natural frequency of the transfer function changes over time, epenent upon the sea state In this case, the N f = 4 fixe moels that we have for the ship are for amping ratio, ζ = 5 an natural frequency, ω n of 1, 125, 15, an 175 ra/s Details are given in the Appenix The isturbance is white noise passe through an integrator The results presente in this section, in Figures 2 to 4, are for a 2 secon simulation where the ship is represente by a secon orer transfer function with amping ratio of 5 which increases in natural frequency ω n begins at 1ra/s an increases by 25ra/s every forty secons until reaching 2ra/s In this way, each of the fixe moel representations is covere plus an extra unknown moel at 2ra/s Probability of 2 is given to each of the fixe moels plus the moel ientifie by recursive least squares The error preiction horizon in (31) is 2 steps an the control horizon is 1 step The weights are Q 1 = 1, Q 2 = 1, R = 1 2 an R 1 = 1 3 The aaptive control scheme is expecte to ientify the moel parameters an tune a PD controller to give an optimal solution across the set Figure 2 epicts the step-reference following capability of the system This is somewhat unrealistic, as in practice the esire ship roll angle is zero egrees However, a square wave input shows the close-loop response more clearly The overshoot an settling time of the 11

12 4 3 Output Reference Signal 2 Ship Angle / Degrees Time / Secs Figure 2: Reference an Ship Roll Angle system varies every 4 secons as woul be expecte from the varying natural frequency Clearly, the ship remains stable with overshoot of no more than 4% Figure 3 shows the 6 ientifie parameters, two from the plant an isturbance moel enominator, a 1 an a 2, two from the plant numerator, b 1 an b 2, an two from the isturbance numerator, c 1 an c 2 It is clear that the a 1 an a 2 values are ecreasing in magnitue over time an the b 1 an b 2 parameters are increasing, ue to the increase in natural frequency The outcome of these parameter variations is given in Figure 4, which shows that both proportional an erivative gains are ecreasing over time Eviently, as the natural frequency an therefore the banwith of the plant rises, it is necessary to ecrease controller gain to avoi instability The system parameter estimates are hel constant at guesse values for the first 6 secons of the simulation until the recursive least squares ata vector is full The aaptive preictive control algorithm upates the PD gains every 4 secons, as it is unnecessary to upate more often for a slowly varying plant For these reasons, control gains are hel constant for the first 8 secons, at which time the algorithm uses the latest ientifie system parameters in the optimisation Clearly, the ientifie system parameters o not approach the correct values until after aroun 25 secons This is an inication that the algorithm is more robust than a stanar self-tuning algorithm that epens upon ientifie parameters only The weight of the 4 fixe moels in the aaptive preictive optimisation keeps the control gains at sensible values, although there is a marke fall immeiately after the algorithm turns on 12

13 System Parameters a1 a2 b1x1 b2x1 c1 c Time / Secs Figure 3: System Parameters 3 K p K 25 2 Controller Gains Time / Secs Figure 4: Controller Gains 13

14 6 Conclusions In this paper, a novel preictive aaptive control technique has been presente The avantage of this metho is the combination of the benefits of self-tuning an multiple-moel restricte-structure optimal controller esigns into one scheme, as well as the incorporation of future set-point knowlege an a multi-step cost inex A self-tuner is able to aapt to changing system parameters at the expense of possible instability A multiple-moel optimal controller gives greater assurance of stability over a wie range of operating points with the expense of conservative performance A multiple-moel aaptive controller is intermeiate to these two schemes It provies a certain amount of confience in stability, ue to the weighte effect of fixe known moels in the optimisation, plus a performance enhancement ue to the incorporation of system ientification knowlege from one sample point to the next The restricte structure of the control law provies simplicity of implementation, an transparency of the solution to those acquainte with much-use classical control laws The preictive aspect of the controller improves setpoint tracking ability an can prouce more efficient use of actuators To further exten an bring rigour to this work, an investigation of the convergence of the restricte structure algorithm woul be esirable Also, a criterion for the robustness of a given multiple-moel problem woul be beneficial Acknowlegement: The authors are grateful for the support of the Engineering an Physical Sciences Research Council (EPSRC) of the UK for their general support via the Platform Grant, Number GR/R4683/1 References [1 J A Richalet, Inustrial applications of moel base preictive control, Automatica, Vol 29, No 8, 1993, [2 C R Cutler an BL Ramaker, Dynamic matrix control - A computer control algorithm, Proceeings JACC, San Francisco, 198 [3 DW Clarke, C Mohtai an PS Tuffs, Generalize preictive control - Part 1, The basic algorithm, Part 2, Extensions an interpretations, Automatica, Vol 23, No 2, 1987, [4 R R Bitmea, M Givers an V Hertz, Optimal control reesign of generalize preictive control, IFAC Symposium on Aaptive Systems in Control an Signal Processing, Glasgow, Scotlan, April 1989 [5 A W Orys an D W Clarke, A state-space escription for GPC controllers, Int J Systems Science, Vol 23, No 2,

15 [6 A W Orys an A W Pike, State-space generalize preictive control incorporating irect through terms, 37th IEEE Control an Decision Conference, Tampa, Floria, 1998 [7 M J Grimble, Two DOF LQG preictive control, IEE Proceeings, Vol 142, No 4, July, 1995, [8 A W Orys an M J Grimble, A multivariable ynamic performance preictive control with application to power generation plants, IFAC Worl Congress, San Francisco, 1996 [9 M J Grimble an A W Orys, Preictive control for inustrial applications, Plenary at IFAC Conference on Control Systems Design, Bratislava, 2 [1 S Li, K Y Lim, D G Fisher, A state space formulation for moel preictive control, AIChE Journal, Vol 35, 1989, [11 V Kucera, Discrete linear control, John Wiley an Sons, Chichester, 1979 [12 M J Grimble, Inustrial Control Systems Design, John Wiley, Chichester,21 [13 M J Grimble, Polynomial solution of preictive optimal control problems for systems in state-equation form, ICC Report 183, University of Strathclye, 21 [14 D G Luenberger, Optimization by Vector Space methos, John Wiley, New York, 1969 [15 N A Hickey, On the Avance Control of Fin Roll Stabilisers in Surface Vessels, PhD Thesis, University of Strathclye, 2 A Appenix A1 Four fixe ship roll moels G ship (s) = θ(s) δ(s) = ω 2 n s 2 + 2ζω n s + w 2 n (A1) θ(s) - Ship Roll Angle, δ(s) - Fin Angle ω n1 = 1, ω n2 = 125, ω n3 = 15, ω n4 = 175 ζ 1 = 1, ζ 2 = 1, ζ 3 = 1, ζ 4 = 1 15