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Approximate model matching with multivariable PI-controllers
Automatica, Vol. 29, No. 6, pp. 1615-1616, 1993
0005-1098/93 $6.00 + (I.110 ~) 1993 Pergamon Press Ltd
Printed in Great Britain.
Technical Communique
Approximate Model Matching with Multivariable Pl-controllers* M I C H A E L K. F. K N O O P t
and J A I M E A. M O R E N O
PEREZ1:
Key Words--Model matching; PI-controller; multivariable systems.
formulae are derived for the gain matrices in Section 2; a stability result is presented in Section 3.
Abstract--#. technique is proposed for the design of multivariable Pl-controllers by approximately matching a prespecified model. The controller parameters are transparently tuned via a scalar model speed parameter. For slow models closed-loop stability can be guaranteed under mild conditions.
2. Controller design For design purposes, a reference step V(s)=~vo is applied to the closed loop. 2.1. Integral gain. The design of KI is based on (2) for t---~0o. Due to the robust properties of Pl-controllers, the steady state error e(oo) vanishes for all stabilizing Kp and K~. Under this condition,
1. Introduction A STRICTLY stable time-invariant multivariable plant, given by its (m, r)-dimensional strictly proper transfer matrix G(s), m and r denoting the size of output y and control input u, respectively. To allow for asymptotic regulation, assume CONSIDER
rk G(O) = m,
y(00) = G(0)u(oo) =
(1)
e(t') dt'
(3)
According to the model matching principle, the closed-loop error is replaced by the model error eM(t ), thus setting equal the error areas of closed loop and model (see Fig. 1). The model error area is obtained as EM(0) from the Laplace
i.e. (a) r -> rn and (b) the plant has no transfer zero at s = 0. The well-known model matching problem seeks for a controller achieving a prespecified closed-loop transfer matrix H(s). This problem has been exactly solved in the state space (Moore and Silverman, 1972), with Wonham's geometric method (Morse, 1973) and by a polynomial matrix approach (Scott and Anderson, 1978). However, these theoretical results are not suited to every technical application, partly because they lead to rather complex controllers, partly because they need a complete and exact model of the plant. On the other hand, engineering approaches not in connection with model matching have been reported for the design of multivariable PI-controllers. Described in the time domain by
u(t) = Kee(t) + K l
G(O)K, , e(t) dt.
domain expression EM(s) = [1,,, - H(s)] ~ v o. So, equation $ (3) reads
v,," y(oo) = G(O)K,.lim 1 [ / ~" . , , - n(s)lv,,. For all v, e R ' , pseudoinverse
it can be solved by a Moore-Penrose K, = - G + ( 0 ) H ' ( 0 ) -t,
(4)
an apostrophe denoting derivation with respect to Laplace variable s. 2.2. Proportional gain. For t--~0+ and under zero initial conditions, equation (2) yields u ( 0 + ) = Kpvo. This control input jump results in a jump
(2)
with control error e = v - y and v being the reference input, Pl-controllers are widely used especially due to their robust properties, i.e. asymptotic tracking and disturbance rejection for constant signals. Rosenbrock (1971) proposed to tune the proportional gain Kp in accordance with the plant's high frequency properties, the integral gain K x with the low frequency properties. Based on this idea, Penttinen and Koivo (1980) and Porter (1982) conceived practical Pl-controller design methods. The aim of the present paper is to combine the model matching idea with the above-mentioned practical design methods. Therefore, exactness in model matching has to be traded for transparency in the design process and for confining the controller structure to Pl-type. Analytical
(d,) Y, (O+ ) : [ ~im ~= sa'g:r(s)]u(O+
= [,~im=sa'g:r(s)]Kpv,,
(5)
of the d~-th derivation of output variable y~, where d~ is its relative degree and glV(s) the i-th row of the plant's transfer matrix G(s). To match the model, the d~-th derivations of closed-loop output and model output should agree. This imposes a restriction on the model transfer matrix H(s), i.e. the i-th model output must have relative degree di, so that
* Received 16 October 1992; recommended for publication by Editor W. S. Levine. t Betriebsforschungsinstitut VDEh, Postfach 1051 45, D-40042 Diisseldorf, Germany. ~:Universitfit der Bundeswehr Hamburg, Fachbereich Elektrotechnik, Postfach 7008 22, D-22008 Hamburg, Germany.
[.~imsaig:r(s)]Kr, v,,=[.~ims'tih:"(s)]v,,
(6)
can be obtained by an appropriate Kp. Fulfilling (6) for i = 1. . . . . m and for all Vo e R " leads to
Kp = G~H o 1615
(7)
1616
Technical Communique go to the left for small a. Their derivation with respect to a is
starting slope
aY~iOa,=o = -wlrG(O)KIoi'
~~i
~J - -
~
i = 1. . . . .
m,
with {vt . . . . . v,,} and {w~, . . . . w~} linear independent and orthogonal, i.e. wirvj = 6ij. These derivations are a times the eigenvalues of H'(0) -~. From this, the proposition follows.
model step response
Remark 1. This proposition is in accordance with the results of Morari (1985).
closed loop step response
Remark 2. The speed parameter a can be used for tuning purposes.
Fit3. 1. Control error area (hatched) and starting slope specified for a plant with relative degree d = 1. with the decoupling matrices Go = l i m
:
and
Ho=lim
l
!
l
of plant and model, respectively. 2.3. Connections to previous work. Porter (1982) derived design equations formally in accordance with (4) and (7). Instead of H'(0) -t and H D, he uses diagonal matrices without direct dynamical motivation, free for tuning purposes. 3. A stability result Let us introduce a speed parameter a ~ R +, slowing down the model H(s/a) for a < 1 and speeding it up for a > 1. For sufficiently slow models, the following proposition guarantees closed loop stablility. Proposition. Consider the strictly stable proper rational plant G(s). There is an a* e R + such that the closed loop with the Pf-controller K, = -aG+(O)H'(O) -I, K e = a G ~ Diag [a dl- i . . . . . a a,,,- I]HD
(8) (9)
is strictly stable for all a e (0, a*], if H'(0) is simple and all its eigenvalues have negative real parts. Proof. For a = 0, both gain matrices are zero, since a strictly proper rational plant has relative degrees di->l. As closed-loop eigenvalues there are eigenvalues of the plant, with negative real parts due to the strict stability, and m eigenvalues in s = 0. Because of the well-known fact that eigenvalues depend continuously on the matrix elements, the plant's eigenvalues remain in the left half-plane for a small enough. It still has to be shown that the m zero eigenvalues
Remark 3. From equations (8) and (9) and the closed-loop characteristic equation, it can be shown that for very fast models, i.e. for large speed parameters a, some of the closed loop eigenvalues tend to the plant's transfer zeros. If the plant is non-minimum phase, the closed loop will then be unstable. 4. Conclusion A method for the design of multivariable Pl-controllers by approximate model matching has been proposed. It allows for a clear dynamical interpretation of the design parameters. The speed parameter a permits a transparent tuning. For slow models, closed-loop stability can be proven under mild conditions on the model. The method needs only steady state and relative degree information on the plant, easily obtainable by off-line experiments. The design method can be extended to sampled-data Pl-controllers and to distributed parameter systems with an appropriate definition of relative degree for non-rational transfer matrices. References Moore, B. C. and L. M. Silverman (1972). Model matching by state feedback and dynamic compensation. I E E E Trans. Aut. Control, 17, 491-497. Morari, M. (1985). Robust stability of systems with integral control. I E E E Trans. Aut. Control, 311, 574-577. Morse, A. S. (1973). Structure and design of linear model following systems. I E E E Trans. Aut. Control, 18, 346-354. Penttinen, J. and H. N. Koivo (1980). Multivariable tuning regulators for unknown plants. Automatica, 16, 393-398. Porter, B. (1982). Design of tunable set-point tracking controllers for linear muitivariable plants. Int. J. Control, 35, 1107-1115. Rosenbrock, H. H. (197l). Progress in the design of multivariable control systems. Measurement and Control, 4, 9-11. Scott, R. W. and B. D. O. Anderson (1978). Least order, stable solution of the exact model matching problem. Automatica, 14, 481-492.
0005-1098/93 $6.00 + (I.110 ~) 1993 Pergamon Press Ltd
Printed in Great Britain.
Technical Communique
Approximate Model Matching with Multivariable Pl-controllers* M I C H A E L K. F. K N O O P t
and J A I M E A. M O R E N O
PEREZ1:
Key Words--Model matching; PI-controller; multivariable systems.
formulae are derived for the gain matrices in Section 2; a stability result is presented in Section 3.
Abstract--#. technique is proposed for the design of multivariable Pl-controllers by approximately matching a prespecified model. The controller parameters are transparently tuned via a scalar model speed parameter. For slow models closed-loop stability can be guaranteed under mild conditions.
2. Controller design For design purposes, a reference step V(s)=~vo is applied to the closed loop. 2.1. Integral gain. The design of KI is based on (2) for t---~0o. Due to the robust properties of Pl-controllers, the steady state error e(oo) vanishes for all stabilizing Kp and K~. Under this condition,
1. Introduction A STRICTLY stable time-invariant multivariable plant, given by its (m, r)-dimensional strictly proper transfer matrix G(s), m and r denoting the size of output y and control input u, respectively. To allow for asymptotic regulation, assume CONSIDER
rk G(O) = m,
y(00) = G(0)u(oo) =
(1)
e(t') dt'
(3)
According to the model matching principle, the closed-loop error is replaced by the model error eM(t ), thus setting equal the error areas of closed loop and model (see Fig. 1). The model error area is obtained as EM(0) from the Laplace
i.e. (a) r -> rn and (b) the plant has no transfer zero at s = 0. The well-known model matching problem seeks for a controller achieving a prespecified closed-loop transfer matrix H(s). This problem has been exactly solved in the state space (Moore and Silverman, 1972), with Wonham's geometric method (Morse, 1973) and by a polynomial matrix approach (Scott and Anderson, 1978). However, these theoretical results are not suited to every technical application, partly because they lead to rather complex controllers, partly because they need a complete and exact model of the plant. On the other hand, engineering approaches not in connection with model matching have been reported for the design of multivariable PI-controllers. Described in the time domain by
u(t) = Kee(t) + K l
G(O)K, , e(t) dt.
domain expression EM(s) = [1,,, - H(s)] ~ v o. So, equation $ (3) reads
v,," y(oo) = G(O)K,.lim 1 [ / ~" . , , - n(s)lv,,. For all v, e R ' , pseudoinverse
it can be solved by a Moore-Penrose K, = - G + ( 0 ) H ' ( 0 ) -t,
(4)
an apostrophe denoting derivation with respect to Laplace variable s. 2.2. Proportional gain. For t--~0+ and under zero initial conditions, equation (2) yields u ( 0 + ) = Kpvo. This control input jump results in a jump
(2)
with control error e = v - y and v being the reference input, Pl-controllers are widely used especially due to their robust properties, i.e. asymptotic tracking and disturbance rejection for constant signals. Rosenbrock (1971) proposed to tune the proportional gain Kp in accordance with the plant's high frequency properties, the integral gain K x with the low frequency properties. Based on this idea, Penttinen and Koivo (1980) and Porter (1982) conceived practical Pl-controller design methods. The aim of the present paper is to combine the model matching idea with the above-mentioned practical design methods. Therefore, exactness in model matching has to be traded for transparency in the design process and for confining the controller structure to Pl-type. Analytical
(d,) Y, (O+ ) : [ ~im ~= sa'g:r(s)]u(O+
= [,~im=sa'g:r(s)]Kpv,,
(5)
of the d~-th derivation of output variable y~, where d~ is its relative degree and glV(s) the i-th row of the plant's transfer matrix G(s). To match the model, the d~-th derivations of closed-loop output and model output should agree. This imposes a restriction on the model transfer matrix H(s), i.e. the i-th model output must have relative degree di, so that
* Received 16 October 1992; recommended for publication by Editor W. S. Levine. t Betriebsforschungsinstitut VDEh, Postfach 1051 45, D-40042 Diisseldorf, Germany. ~:Universitfit der Bundeswehr Hamburg, Fachbereich Elektrotechnik, Postfach 7008 22, D-22008 Hamburg, Germany.
[.~imsaig:r(s)]Kr, v,,=[.~ims'tih:"(s)]v,,
(6)
can be obtained by an appropriate Kp. Fulfilling (6) for i = 1. . . . . m and for all Vo e R " leads to
Kp = G~H o 1615
(7)
1616
Technical Communique go to the left for small a. Their derivation with respect to a is
starting slope
aY~iOa,=o = -wlrG(O)KIoi'
~~i
~J - -
~
i = 1. . . . .
m,
with {vt . . . . . v,,} and {w~, . . . . w~} linear independent and orthogonal, i.e. wirvj = 6ij. These derivations are a times the eigenvalues of H'(0) -~. From this, the proposition follows.
model step response
Remark 1. This proposition is in accordance with the results of Morari (1985).
closed loop step response
Remark 2. The speed parameter a can be used for tuning purposes.
Fit3. 1. Control error area (hatched) and starting slope specified for a plant with relative degree d = 1. with the decoupling matrices Go = l i m
:
and
Ho=lim
l
!
l
of plant and model, respectively. 2.3. Connections to previous work. Porter (1982) derived design equations formally in accordance with (4) and (7). Instead of H'(0) -t and H D, he uses diagonal matrices without direct dynamical motivation, free for tuning purposes. 3. A stability result Let us introduce a speed parameter a ~ R +, slowing down the model H(s/a) for a < 1 and speeding it up for a > 1. For sufficiently slow models, the following proposition guarantees closed loop stablility. Proposition. Consider the strictly stable proper rational plant G(s). There is an a* e R + such that the closed loop with the Pf-controller K, = -aG+(O)H'(O) -I, K e = a G ~ Diag [a dl- i . . . . . a a,,,- I]HD
(8) (9)
is strictly stable for all a e (0, a*], if H'(0) is simple and all its eigenvalues have negative real parts. Proof. For a = 0, both gain matrices are zero, since a strictly proper rational plant has relative degrees di->l. As closed-loop eigenvalues there are eigenvalues of the plant, with negative real parts due to the strict stability, and m eigenvalues in s = 0. Because of the well-known fact that eigenvalues depend continuously on the matrix elements, the plant's eigenvalues remain in the left half-plane for a small enough. It still has to be shown that the m zero eigenvalues
Remark 3. From equations (8) and (9) and the closed-loop characteristic equation, it can be shown that for very fast models, i.e. for large speed parameters a, some of the closed loop eigenvalues tend to the plant's transfer zeros. If the plant is non-minimum phase, the closed loop will then be unstable. 4. Conclusion A method for the design of multivariable Pl-controllers by approximate model matching has been proposed. It allows for a clear dynamical interpretation of the design parameters. The speed parameter a permits a transparent tuning. For slow models, closed-loop stability can be proven under mild conditions on the model. The method needs only steady state and relative degree information on the plant, easily obtainable by off-line experiments. The design method can be extended to sampled-data Pl-controllers and to distributed parameter systems with an appropriate definition of relative degree for non-rational transfer matrices. References Moore, B. C. and L. M. Silverman (1972). Model matching by state feedback and dynamic compensation. I E E E Trans. Aut. Control, 17, 491-497. Morari, M. (1985). Robust stability of systems with integral control. I E E E Trans. Aut. Control, 311, 574-577. Morse, A. S. (1973). Structure and design of linear model following systems. I E E E Trans. Aut. Control, 18, 346-354. Penttinen, J. and H. N. Koivo (1980). Multivariable tuning regulators for unknown plants. Automatica, 16, 393-398. Porter, B. (1982). Design of tunable set-point tracking controllers for linear muitivariable plants. Int. J. Control, 35, 1107-1115. Rosenbrock, H. H. (197l). Progress in the design of multivariable control systems. Measurement and Control, 4, 9-11. Scott, R. W. and B. D. O. Anderson (1978). Least order, stable solution of the exact model matching problem. Automatica, 14, 481-492.