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Robust feedback synthesis for margins at the plant input
Automntica,
Pergamon
Vol. 31, No. 2, pp. 333-336, 1995 1995 Elsevier Science Ltd in Great Britain. All rights reserved ooos109w95 S9.M + 0.00
Copflgh’0
ooo5-1098(94)00104-9
Printed
Brief Paper
Robust Feedback Synthesis for Margins at the Plant Input* ODED YANIVt Key Words-Control frequency domain.
theory;
feedback
control;
multivariable
control
systems;
robust
control;
stability, assuming memoryless elements at each one of the plant inputs at a time, for example, when a plant input is saturated. The paper is set out as follows: the problem is defined in Section 2, while Section 3 develops the design process, Section 4 discusses the conditions for a successful execution of the design process, Section 5 presents a design example, and the conclusions are given in Section 6.
Abstract-In its present form, the Quantitative Feedback Theory (QFT) for uncertain MIMO feedback systems is a tool to design the two degree-of-freedom of a feedback system. It satisfies demands of robustness, performance, and gain and phase margin at the plant input for diagonal controllers. The method is now extended to meet the requirements of gain and phase margin specifications (including as a function of frequency) at the plant input for non-diagonal controllers. The advantages are: (1) elimination of underdamped closed loop transfer function from any plant input into itself and elimination of underdamped closed loop poles in the effective bandwidth of the system. In the most general terms the time response at the plant input owing to disturbances, sensor noise and/or tracking commands is improved and long-duration ‘ringing’ is avoided; (2) the margins fit the conditions of the circle criterion of guarantee stability for memoryless elements at one of the plant inputs; and (3) the design can be tailored to meet disturbance rejection specifications at the plant input for a given disturbance spectrum.
2. Definidon of rhe problem Consider the feedback system shown in Fig. 1, where the uncertain plant P = [p,]$ belongs to the family 9. The controller WG is a product of the pre-controller W and the diagonal controller G. The QFT technique to design W, G and F is a two-step process. First W is designed such that PW is a square matrix (Horowitz, 1991), and the uncertain plant set 5iW has properties which are to some extent superior to those of plant set 9! Then the QFT technique (Yaniv and Horowitz, 1986) and its extension (Yaniv, 1992) are used to design the diagonal controller G and prefilter F. Let the following definitions hold: n, m, number of rows (outputs) and columns (inputs) of P. % the frequency above which sensitivity to plant variations can be neglected. ail(o), bij(o) and cij, sensitivity specs, i, j = 1, . , n. d,(w), margin specs; there exist o1 and xi such that di(w)>xj>l for o?o,, i=l,..., m. [t,], the transfer functifn from r to y (Fig. 1). ;)[d,] the transfer function from the plant output to y (Fig.
1. Introduction The robust synthesis
l
of MIMO feedback systems is a problem which has aroused a great deal of interest. Some of the most important synth?sis methods are H, (Francis, 1987), Adaptive Control (Astriim, 1985), the techniques of Youla and Bongiomo (1985), the British school (Rosenbrock, 19774;McFarlane 1977; Mayne, 1979), and the QFT method (Yaniv and Horowitz, 1986, Yaniv, 1992). In addition to robust stability and performance, a good synthesis technique should meet the demands of gain and phase margin at the plant input. This subject has been discussed in several papers, using various definitions of gain and phase margin (for example, Sobel et al., 1983; Davison, 1986; Tannenbaum, 1986; Yaniv, 1992). In the present study the definition of Yaniv (1992) is adopted and extended to margins at the plant input. Its advantages are: (1) it is a natural extension of the SISO approach; hence a smooth (not underdamped) transfer function from any plant input into itself is guaranteed, long-duration ‘ringing’ is avoided and a short settle time can be achieved, (2) the margins can be stated as a function of frequency, so that disturbance attenuation at the plant input can also be a function of frequency and so be tailored to the disturbance spectrum; and (3) it meets the circle criterion for guaranteeing
l
l
l
l
Li, the loop transmission from plant input i into itself, derived as follows: disconnect the plant input i from output i of W. Then inject an impulse into the plant input i and measure the signal returned at the ith output of W. The transfer function of the latter is Li. l
Remark 2.1. The frequencies o0 and o1 exist in all strictly proper systems because at high frequencies the benefits of feedback are negligible.
Statement of the problem: given W, find F and G such that for all P E 9, the system defined in Fig. 1 is internally stable, and for all i and j: Closed loop performance: for given o,,, aij, b, and cij, one or both of the following: l
aij(o) = Itij(iw)l = b,(w);
*Received 23 March 1993; revised 2 November 1993; received in final form 17 May 1994. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Eugenius Kaszkurewicz under the direction of editor Huibert Kwakemaak. Corresponding author Dr Oded Yaniv. Tel. +972 3 6408764; Fax +972 3 6407095; E-mail [email protected]. t Faculty of Engineering, Department of Electrical Engineering Systems, Tel Aviv University, Tel Aviv 69 978, Israel.
Idij(iw)l 5 Cij(o);
Margin performance: d,(o) 1
I I
sdi(o),
i=l,...,
$ Lower case italic letters denote capital letters P, W, G and F. 333
(1)
for a given margin performance
l
__ l+L{
OSOrJ 0 5 00.
m. the elements
(2) of the
334
Brief Papers Then
Fig. I. A MIMO two DOF feedback structure. From equations (8) and (4) the margin performance given k Note that L, is a transfer function of a SISO system; hence if the output i of W is connected to the ith input of P. the ith plant input owing to disturbance at the ith plant input will be __ 1 ZZji 1 + L,
element of
[I + WGP] ‘.
(3)
Hence margin performance (2) can be replaced by: Margin performance: for given margin performance d,(w) l
[the ii element of (I + WGP] -‘I s.d,(w).
i = 1..
, m. (4)
That is, the ith plant input to disturbance at the ith plant input will not be amplified by more than d,(w). For example if d, = 2 for all frequencies then 30 deg and 6 dB margins are guaranteed which eliminates underdamped loop transmissions L,. A solution to this problem, for square plants and diagonal controllers, was presented by Yaniv (1992). But it is more likely that controllers will not be diagonal and plants not square. The Yaniv (1992) technique can be used to solve the problem as follows: add to the ith plant input the multiplication uncertainty I/(I + A,).
(5)
IA,/= l/d,(o).
where iA,/< 1. Then a stable solution will be guaranteed if
ll + L,l > 141= l/d,(w).
(7)
The drawbacks of this technique are that it works only for IA,/< 1, that it includes overdesign, and that adding unstructured uncertainty increases the calculation efforts to an unrealistic level. For example, suppose that each multiplication uncertainty is modeled by 20 cases and n = 3. the number of plant cases will increase by 203. The next section gives an algorithm for non-diagonal controllers and non-square plants that overcomes these drawbacks. 3. Development of the design procedure The QFT design method (Yaniv and Horowitz, 1986; Yaniv. 1992) turns the design process into a sequence of SISO problems. The solution of each SISO problem is the controller g, and the prefilters A,, j = 1,. , n (each A, being due to the closed loop specifications from input at j to output at i), giving a combined solution G = diag (g,) and F = [A,] for the MIMO system. The main task during the design of each SISO problem is to find bounds on an open loop transfer function. Each bound is a closed curve on the complex plane, dividing it into two regions, one of which is called u(w), so that if the open loop transfer function at frequency w belongs to (T(O) and satisfies the Nyquist stability criterion, the synthesis procedure must succeed. The design process becomes a kind of simultaneous stabilization problem with constraints; that is, at each step g, is designed so that at each frequency w, it lies in its permissible region (T(W)and stabilizes a given class of plants. Returning to the problem defined in Section 2. the question is what other constraints have to be imposed on each design step so that the margin performance will be satisfied. In order to establish an algorithm, the following equations are provided (for a proof see Appendix): let . . . . . O,g,,O ,...,
’
i = 1,.
, m,
(9)
where k = 1,. , n and of, /3!, yf are the diagonal i elements of the corresponding matrices in equation (8). Equation (9) includes m inequalities for the parameter gK. Its solution is a domain ~~(0) that can easily be calculated, because for an assumed phase of gk it is a quadratic inequality on its amplitude; i.e. the basic procedure of QFT calculation of bounds is used. This design algorithm follows: In each of the MIMO QFT design steps, g, should lie in the intersection of the domains a(o) and ok(w). Since the MIMO QFI is a series process (g, is designed in the kth step), in order to calculate Us an assumption as to gi for i > k should be made. Two options are proposed: (1) all g, = =; and (2) as in Yaniv (1992) each one of gk+,, . ,g, can be 0 or =. The following is a simple formula to calculate at. fit and yf where it is known that some of the g,s are 0 and/or =. Let G” = diag (gk) be a diagonal controller whose diagonal elements (Y= (Y,, , a, are zero, G;’ a diagonal matrix whose diagonal elements a are 1 and the elements (Y,, i > I are 1/g,,, P, the same as P but the o rows are zero, and W,, the same as W but the Q columns are zero. Then + G,‘]-‘P,.
(IO)
0).
For a proof, see Appendix. 3.1. Memoryless elements. We shall now show how to guarantee stability if a single memoryless system is introduced at one of the plant inputs. This may happen, for example, when one of the actuators is saturated. Suppose N is a memoryless element at plant input q, and there exists a constant S such that (l-S)ucNui(l+6)u,
(II)
then the circle criterion (Desoer and Vidyasagar, 1975) guarantees stability if the system is stable for N = 1 and
This objective can be achieved by replacing margin performance (2) for channel 4 by inequality (12). From the trivial equation L/( 1 + L) + l/( 1 + L) = 1 the modification of inequalities (9) for a bounds calculation to satisfy inequality (12) will be:
’-
G(w)+gk(dxW) - P:(w))< 1 + &%Y@J)
--
6
k
=
r-
-30
I
~35”
1
,
.,
, n.
(13)
I w-2 3----------. .... ..
20
?OL and
(o)
’
(6)
because if inequality 6 is true for all IA,1= l/d, then
c, = I + W[G - GI,]P.
1 +gkyf(w)
[I + WG”P]-’ = I - W,[P,W,
11+ L,l(l + A\,)/>().
G,=diag(O
Ia:(w) + &P:(w)
is: for
I -300
-250
-200
-150
-100
PHASE
Fig. 2. Bounds and open loop for L,.
-50
0
Brief Papers Find the domains o*(o) on gz, using the QFI MIMO technique of Yaniv (1992). Then find oz(o) to satisfy margin performance (2), using inequality (9) for k = 2, i = 1, . , m, the known g, and g, = 00, where I= 3,. . . , n. Then shape ga(s) so as to satisfy the bounds of the intersection of both domains, and encircle the domain d(o) to satisfy the Nyquist criterion. (3) For the kth step: find the domains o’(o) on g*, using the QFI MIMO technique of Yaniv (1992). Then find ok(o) to satisfy margin performance (2), using inequality (9) for k, i = 1,. . . , m, the known gk_, and g, = m, where 1= k + 1, . , n. Then shape g,,..., g,Js) so as to satisfy the bounds of the intersection of both domains, and encircle the domain o“(w) to satisfy the Nyquist criterion. Clearly, a successful execution of the design process is assured if: (1) The bounds on gi at low frequencies are such that large g, are allowed. From inequality (9) the condition is
PHASE VI= 1.oooo w=z.oMM .. .. ..
w=3.OcQLl w’~‘j:oilQO - _.- - -
VI= 10.OOuO w = 2o.oooO
w=4o.oOoO
Ipro I Y:(o)
Fig. 3. Bounds and open loop for LT.
4. Discussion and suflcient conditions For square plants lim [I + WG”P]-’ = 0. g,~~,i=l,.._,n
(14)
That is, if the uncertain plant is an arbitrarily small sensitivity plant (a plant is called arbitrarily small sensitivity if there exist a solution to any QFI closed loop performance. Conditions for an uncertain plant to be an arbitrarily small sensitivity plant were given by Yaniv, (1991)) any margin performance on any finite frequency range can be achieved. If P has more inputs than outputs lim [I + WG”P]-’ = I - W[PW]-‘P # [O]. (15) s,+=,i=r ,...,fl Thus limitations on the margin performance di(o) are to be expected. But the main limitations can be explained if the designer goes through the following procedure for a n output m input system: the design procedure is the next n steps: (1) Find the domains a’(w) on g,, using the QFT MIMO technique of Yaniv (1992). Then find a,(w) to satisfy margin performance (2), using inequality (9) for k = 1, i = 1,. . . , m and g,=m, where I = 2, . , n. Then shape g,(s) so as to satisfy the bounds of the intersection of both domains, and encircle the domain a’(w) to satisfy the Nyquist criterion. (2)
20
i=l,...,
m.
(2) There exist o. such that the bounds for o > o,, are closed curves in the complex plane that allow a negative finite phase for g,. Clearly a lower allowed phase for g, will decrease its bandwidth. Under what conditions can a solution exist? The answer depends on the set B and both performances. If margin performance (2) is ignored, a partial answer can be found (Yaniv, 1991; Nwokah er al., 1993). The above criteria can serve as guidelines for research on this topic. 5. An example
In Fig. 1 the uncertain plant family is
P=;[;::
;;:I,
(17)
where k,,, kz2 E [2,4], and k12, kZ1 E [l, 1.81, and WC
l [ -0.7
-0.7 1
1
The closed loop specifications 1 are: at o = 1,2,3 the transfer function from the plant output to y will be, in both channels, less than 0.2,0.4 and 2, respectively. The closed loop specifications 2 are: for o I 5, d,(w) = 2; for o = 10, di(o) = 2.5; for o = 20, d,(o) = 3; and for o 2 40, di(o) = 4. 5.1. Design. Figure 2 shows the bounds which solve
Plt Innut to Pit IrIP. 1
;
1
:
(18)
11
-80’ 0
’ 0.5
1 log w
Fig. 4. Simulation to validate closed loop specifications.
1.5
I 2
Brief Papers Inequality 9 for k = 1, assuming g, = z, and which intersect with the usual QFT bounds that guarantee specifications 1. Figure 2 also presents the open loop for the nominal plant. The controller is 18 g, =-
1 + s/50
Figure 3 shows the bounds which solve inequalities 9 for k = 2 and the designed g,, and which intersect with the usual QFT bounds that guarantee specifications 1. Figure 3 also presents the open loop for the nominal plant. The controller is
Horowitz, I. (1991). A survey of quantitative feedback theory (QFT). ht. J. Contr., 53, 255-291. Kailath, T. (1980). Linear Systems. Prentice-Hall, Engelwood Cliffs, N.J. McFarlane, G. J. (1977). Frequency-Response Methods in Control Systems. IEEE Press, John Wiley & Sons, New York. Mayne, D. Q. (1979). Sequential design of linear multivariable svstems. In Proc. IEE. 126.568-572. Nwokah, 0. D. I., D. F. Thompson and R. A. Perez (1993). Almost decoupling by quantitative feedback theory. Trans. ASME J. Dynamic Systems Measurement and Control, 115,
27-37.
g* =13
1 +s/18’
5.2. Simulations. Simulations that validate Performance 2 are presented in Fig. 4. In Fig. 3 the open loop touches the bounds at o = 20, so that no overdesign should be expected. This is shown in the simulation in Fig. 4, where the its maximum is about 8dB which is about the specification on d, at that frequency. This guides the designer in how to eliminate overdesign, i.e. by shaping the controller to be as close as possible to the bounds. 6. Conclusions The correlation between gain and phase margin and a system’s time response is well-known in SISO systems. In this paper, the definition of the SISO gain and phase margin is extended to the plant input of a MIMO system with non-diagonal controllers. The QFT design procedure for uncertain MIMO feedback systems has been extended in order to satisfy gain and phase margin performance (which can be a function of w) at the plant input. Thus the transfer function from each plant input into itself will have the features of a SISO system with the same gain and phase margins. Compact equations for bounds calculated at each design step are presented. References Astrom, K. J. (1985). Adaptive control-a way to deal with uncertainty. In J. Ackerman (Ed.), Uncertainty and Control Lecture Notes in Control and Information Science, p. 70, Springer, Berlin. Davison. E. J. (1986). Robust control for industrial svstems. In P&c. of 2h Ck, WPI, Athens, Greece. . Desoer, C. and M. Vidyasagar (1975). Feedback Systems: Input-Output Properties. Academic Press, New York. Francis, A. B. (1987). A course in H, control theory. In Lecture Notes in Control and Information Sciences. p. 88. Springer, Berlin.
Rosenbrock, H. H. (1974). Computer Aided Control System Design. Academic Press, London. Sobel, K. M., J. C. Chung and E. Y. Shapiro (1983). Application of MIMO phase and gain margins to the evaluation of a flight control system. In Proc. ACC. 3, 1286-1287. Tannenbaum, A. (1986). On the multivariable gain margin problem. Automatica, 22, 381-383. Yaniv. 0. (1991). Arbitrarily small sensitivity in multipleinput-output uncertain feedback systems. Automatica, 27, 565-568. Yaniv. 0. (1992). Synthesis of uncertain MIMO feedback systems for gain and phase margin at different channel breaking points. Automatica, 28, 1017-1020 Yaniv. 0. and I. Horowitz (1986). A quantitative design method for MIMO linear feedback systems having uncertain plants. ht. J. Contr., 43, 401-421. Youla, D. C. and J. J. Bongiorno (1985). A feedback theory of two degree-of-freedom optimal Wiener-Hopf design. IEEE Trans. Autom. Contr., AC-SO, 652-665. Appendix Proof of equafion (8). Trivial consequence p. 655), applied on the right side of (I+WGP]
of Kailath
‘=[l+W(GpG,)P)+WG,P]-’ = [C + g,uu’]
‘.
Proof of equation (10). Since rows and columns zero and G” is diagonal I + WG”P
(1980,
= I + W,“G”P,
= I + W,G,P,,
(A.1) (Yof G” are
(A.2)
where G, is the same as G, except for the diagonal zero elements, which are replaced by 1. Now use Kailath (1980, p. 656) and substitute G,’ = G, ‘.
Pergamon
Vol. 31, No. 2, pp. 333-336, 1995 1995 Elsevier Science Ltd in Great Britain. All rights reserved ooos109w95 S9.M + 0.00
Copflgh’0
ooo5-1098(94)00104-9
Printed
Brief Paper
Robust Feedback Synthesis for Margins at the Plant Input* ODED YANIVt Key Words-Control frequency domain.
theory;
feedback
control;
multivariable
control
systems;
robust
control;
stability, assuming memoryless elements at each one of the plant inputs at a time, for example, when a plant input is saturated. The paper is set out as follows: the problem is defined in Section 2, while Section 3 develops the design process, Section 4 discusses the conditions for a successful execution of the design process, Section 5 presents a design example, and the conclusions are given in Section 6.
Abstract-In its present form, the Quantitative Feedback Theory (QFT) for uncertain MIMO feedback systems is a tool to design the two degree-of-freedom of a feedback system. It satisfies demands of robustness, performance, and gain and phase margin at the plant input for diagonal controllers. The method is now extended to meet the requirements of gain and phase margin specifications (including as a function of frequency) at the plant input for non-diagonal controllers. The advantages are: (1) elimination of underdamped closed loop transfer function from any plant input into itself and elimination of underdamped closed loop poles in the effective bandwidth of the system. In the most general terms the time response at the plant input owing to disturbances, sensor noise and/or tracking commands is improved and long-duration ‘ringing’ is avoided; (2) the margins fit the conditions of the circle criterion of guarantee stability for memoryless elements at one of the plant inputs; and (3) the design can be tailored to meet disturbance rejection specifications at the plant input for a given disturbance spectrum.
2. Definidon of rhe problem Consider the feedback system shown in Fig. 1, where the uncertain plant P = [p,]$ belongs to the family 9. The controller WG is a product of the pre-controller W and the diagonal controller G. The QFT technique to design W, G and F is a two-step process. First W is designed such that PW is a square matrix (Horowitz, 1991), and the uncertain plant set 5iW has properties which are to some extent superior to those of plant set 9! Then the QFT technique (Yaniv and Horowitz, 1986) and its extension (Yaniv, 1992) are used to design the diagonal controller G and prefilter F. Let the following definitions hold: n, m, number of rows (outputs) and columns (inputs) of P. % the frequency above which sensitivity to plant variations can be neglected. ail(o), bij(o) and cij, sensitivity specs, i, j = 1, . , n. d,(w), margin specs; there exist o1 and xi such that di(w)>xj>l for o?o,, i=l,..., m. [t,], the transfer functifn from r to y (Fig. 1). ;)[d,] the transfer function from the plant output to y (Fig.
1. Introduction The robust synthesis
l
of MIMO feedback systems is a problem which has aroused a great deal of interest. Some of the most important synth?sis methods are H, (Francis, 1987), Adaptive Control (Astriim, 1985), the techniques of Youla and Bongiomo (1985), the British school (Rosenbrock, 19774;McFarlane 1977; Mayne, 1979), and the QFT method (Yaniv and Horowitz, 1986, Yaniv, 1992). In addition to robust stability and performance, a good synthesis technique should meet the demands of gain and phase margin at the plant input. This subject has been discussed in several papers, using various definitions of gain and phase margin (for example, Sobel et al., 1983; Davison, 1986; Tannenbaum, 1986; Yaniv, 1992). In the present study the definition of Yaniv (1992) is adopted and extended to margins at the plant input. Its advantages are: (1) it is a natural extension of the SISO approach; hence a smooth (not underdamped) transfer function from any plant input into itself is guaranteed, long-duration ‘ringing’ is avoided and a short settle time can be achieved, (2) the margins can be stated as a function of frequency, so that disturbance attenuation at the plant input can also be a function of frequency and so be tailored to the disturbance spectrum; and (3) it meets the circle criterion for guaranteeing
l
l
l
l
Li, the loop transmission from plant input i into itself, derived as follows: disconnect the plant input i from output i of W. Then inject an impulse into the plant input i and measure the signal returned at the ith output of W. The transfer function of the latter is Li. l
Remark 2.1. The frequencies o0 and o1 exist in all strictly proper systems because at high frequencies the benefits of feedback are negligible.
Statement of the problem: given W, find F and G such that for all P E 9, the system defined in Fig. 1 is internally stable, and for all i and j: Closed loop performance: for given o,,, aij, b, and cij, one or both of the following: l
aij(o) = Itij(iw)l = b,(w);
*Received 23 March 1993; revised 2 November 1993; received in final form 17 May 1994. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Eugenius Kaszkurewicz under the direction of editor Huibert Kwakemaak. Corresponding author Dr Oded Yaniv. Tel. +972 3 6408764; Fax +972 3 6407095; E-mail [email protected]. t Faculty of Engineering, Department of Electrical Engineering Systems, Tel Aviv University, Tel Aviv 69 978, Israel.
Idij(iw)l 5 Cij(o);
Margin performance: d,(o) 1
I I
sdi(o),
i=l,...,
$ Lower case italic letters denote capital letters P, W, G and F. 333
(1)
for a given margin performance
l
__ l+L{
OSOrJ 0 5 00.
m. the elements
(2) of the
334
Brief Papers Then
Fig. I. A MIMO two DOF feedback structure. From equations (8) and (4) the margin performance given k Note that L, is a transfer function of a SISO system; hence if the output i of W is connected to the ith input of P. the ith plant input owing to disturbance at the ith plant input will be __ 1 ZZji 1 + L,
element of
[I + WGP] ‘.
(3)
Hence margin performance (2) can be replaced by: Margin performance: for given margin performance d,(w) l
[the ii element of (I + WGP] -‘I s.d,(w).
i = 1..
, m. (4)
That is, the ith plant input to disturbance at the ith plant input will not be amplified by more than d,(w). For example if d, = 2 for all frequencies then 30 deg and 6 dB margins are guaranteed which eliminates underdamped loop transmissions L,. A solution to this problem, for square plants and diagonal controllers, was presented by Yaniv (1992). But it is more likely that controllers will not be diagonal and plants not square. The Yaniv (1992) technique can be used to solve the problem as follows: add to the ith plant input the multiplication uncertainty I/(I + A,).
(5)
IA,/= l/d,(o).
where iA,/< 1. Then a stable solution will be guaranteed if
ll + L,l > 141= l/d,(w).
(7)
The drawbacks of this technique are that it works only for IA,/< 1, that it includes overdesign, and that adding unstructured uncertainty increases the calculation efforts to an unrealistic level. For example, suppose that each multiplication uncertainty is modeled by 20 cases and n = 3. the number of plant cases will increase by 203. The next section gives an algorithm for non-diagonal controllers and non-square plants that overcomes these drawbacks. 3. Development of the design procedure The QFT design method (Yaniv and Horowitz, 1986; Yaniv. 1992) turns the design process into a sequence of SISO problems. The solution of each SISO problem is the controller g, and the prefilters A,, j = 1,. , n (each A, being due to the closed loop specifications from input at j to output at i), giving a combined solution G = diag (g,) and F = [A,] for the MIMO system. The main task during the design of each SISO problem is to find bounds on an open loop transfer function. Each bound is a closed curve on the complex plane, dividing it into two regions, one of which is called u(w), so that if the open loop transfer function at frequency w belongs to (T(O) and satisfies the Nyquist stability criterion, the synthesis procedure must succeed. The design process becomes a kind of simultaneous stabilization problem with constraints; that is, at each step g, is designed so that at each frequency w, it lies in its permissible region (T(W)and stabilizes a given class of plants. Returning to the problem defined in Section 2. the question is what other constraints have to be imposed on each design step so that the margin performance will be satisfied. In order to establish an algorithm, the following equations are provided (for a proof see Appendix): let . . . . . O,g,,O ,...,
’
i = 1,.
, m,
(9)
where k = 1,. , n and of, /3!, yf are the diagonal i elements of the corresponding matrices in equation (8). Equation (9) includes m inequalities for the parameter gK. Its solution is a domain ~~(0) that can easily be calculated, because for an assumed phase of gk it is a quadratic inequality on its amplitude; i.e. the basic procedure of QFT calculation of bounds is used. This design algorithm follows: In each of the MIMO QFT design steps, g, should lie in the intersection of the domains a(o) and ok(w). Since the MIMO QFI is a series process (g, is designed in the kth step), in order to calculate Us an assumption as to gi for i > k should be made. Two options are proposed: (1) all g, = =; and (2) as in Yaniv (1992) each one of gk+,, . ,g, can be 0 or =. The following is a simple formula to calculate at. fit and yf where it is known that some of the g,s are 0 and/or =. Let G” = diag (gk) be a diagonal controller whose diagonal elements (Y= (Y,, , a, are zero, G;’ a diagonal matrix whose diagonal elements a are 1 and the elements (Y,, i > I are 1/g,,, P, the same as P but the o rows are zero, and W,, the same as W but the Q columns are zero. Then + G,‘]-‘P,.
(IO)
0).
For a proof, see Appendix. 3.1. Memoryless elements. We shall now show how to guarantee stability if a single memoryless system is introduced at one of the plant inputs. This may happen, for example, when one of the actuators is saturated. Suppose N is a memoryless element at plant input q, and there exists a constant S such that (l-S)ucNui(l+6)u,
(II)
then the circle criterion (Desoer and Vidyasagar, 1975) guarantees stability if the system is stable for N = 1 and
This objective can be achieved by replacing margin performance (2) for channel 4 by inequality (12). From the trivial equation L/( 1 + L) + l/( 1 + L) = 1 the modification of inequalities (9) for a bounds calculation to satisfy inequality (12) will be:
’-
G(w)+gk(dxW) - P:(w))< 1 + &%Y@J)
--
6
k
=
r-
-30
I
~35”
1
,
.,
, n.
(13)
I w-2 3----------. .... ..
20
?OL and
(o)
’
(6)
because if inequality 6 is true for all IA,1= l/d, then
c, = I + W[G - GI,]P.
1 +gkyf(w)
[I + WG”P]-’ = I - W,[P,W,
11+ L,l(l + A\,)/>().
G,=diag(O
Ia:(w) + &P:(w)
is: for
I -300
-250
-200
-150
-100
PHASE
Fig. 2. Bounds and open loop for L,.
-50
0
Brief Papers Find the domains o*(o) on gz, using the QFI MIMO technique of Yaniv (1992). Then find oz(o) to satisfy margin performance (2), using inequality (9) for k = 2, i = 1, . , m, the known g, and g, = 00, where I= 3,. . . , n. Then shape ga(s) so as to satisfy the bounds of the intersection of both domains, and encircle the domain d(o) to satisfy the Nyquist criterion. (3) For the kth step: find the domains o’(o) on g*, using the QFI MIMO technique of Yaniv (1992). Then find ok(o) to satisfy margin performance (2), using inequality (9) for k, i = 1,. . . , m, the known gk_, and g, = m, where 1= k + 1, . , n. Then shape g,,..., g,Js) so as to satisfy the bounds of the intersection of both domains, and encircle the domain o“(w) to satisfy the Nyquist criterion. Clearly, a successful execution of the design process is assured if: (1) The bounds on gi at low frequencies are such that large g, are allowed. From inequality (9) the condition is
PHASE VI= 1.oooo w=z.oMM .. .. ..
w=3.OcQLl w’~‘j:oilQO - _.- - -
VI= 10.OOuO w = 2o.oooO
w=4o.oOoO
Ipro I Y:(o)
Fig. 3. Bounds and open loop for LT.
4. Discussion and suflcient conditions For square plants lim [I + WG”P]-’ = 0. g,~~,i=l,.._,n
(14)
That is, if the uncertain plant is an arbitrarily small sensitivity plant (a plant is called arbitrarily small sensitivity if there exist a solution to any QFI closed loop performance. Conditions for an uncertain plant to be an arbitrarily small sensitivity plant were given by Yaniv, (1991)) any margin performance on any finite frequency range can be achieved. If P has more inputs than outputs lim [I + WG”P]-’ = I - W[PW]-‘P # [O]. (15) s,+=,i=r ,...,fl Thus limitations on the margin performance di(o) are to be expected. But the main limitations can be explained if the designer goes through the following procedure for a n output m input system: the design procedure is the next n steps: (1) Find the domains a’(w) on g,, using the QFT MIMO technique of Yaniv (1992). Then find a,(w) to satisfy margin performance (2), using inequality (9) for k = 1, i = 1,. . . , m and g,=m, where I = 2, . , n. Then shape g,(s) so as to satisfy the bounds of the intersection of both domains, and encircle the domain a’(w) to satisfy the Nyquist criterion. (2)
20
i=l,...,
m.
(2) There exist o. such that the bounds for o > o,, are closed curves in the complex plane that allow a negative finite phase for g,. Clearly a lower allowed phase for g, will decrease its bandwidth. Under what conditions can a solution exist? The answer depends on the set B and both performances. If margin performance (2) is ignored, a partial answer can be found (Yaniv, 1991; Nwokah er al., 1993). The above criteria can serve as guidelines for research on this topic. 5. An example
In Fig. 1 the uncertain plant family is
P=;[;::
;;:I,
(17)
where k,,, kz2 E [2,4], and k12, kZ1 E [l, 1.81, and WC
l [ -0.7
-0.7 1
1
The closed loop specifications 1 are: at o = 1,2,3 the transfer function from the plant output to y will be, in both channels, less than 0.2,0.4 and 2, respectively. The closed loop specifications 2 are: for o I 5, d,(w) = 2; for o = 10, di(o) = 2.5; for o = 20, d,(o) = 3; and for o 2 40, di(o) = 4. 5.1. Design. Figure 2 shows the bounds which solve
Plt Innut to Pit IrIP. 1
;
1
:
(18)
11
-80’ 0
’ 0.5
1 log w
Fig. 4. Simulation to validate closed loop specifications.
1.5
I 2
Brief Papers Inequality 9 for k = 1, assuming g, = z, and which intersect with the usual QFT bounds that guarantee specifications 1. Figure 2 also presents the open loop for the nominal plant. The controller is 18 g, =-
1 + s/50
Figure 3 shows the bounds which solve inequalities 9 for k = 2 and the designed g,, and which intersect with the usual QFT bounds that guarantee specifications 1. Figure 3 also presents the open loop for the nominal plant. The controller is
Horowitz, I. (1991). A survey of quantitative feedback theory (QFT). ht. J. Contr., 53, 255-291. Kailath, T. (1980). Linear Systems. Prentice-Hall, Engelwood Cliffs, N.J. McFarlane, G. J. (1977). Frequency-Response Methods in Control Systems. IEEE Press, John Wiley & Sons, New York. Mayne, D. Q. (1979). Sequential design of linear multivariable svstems. In Proc. IEE. 126.568-572. Nwokah, 0. D. I., D. F. Thompson and R. A. Perez (1993). Almost decoupling by quantitative feedback theory. Trans. ASME J. Dynamic Systems Measurement and Control, 115,
27-37.
g* =13
1 +s/18’
5.2. Simulations. Simulations that validate Performance 2 are presented in Fig. 4. In Fig. 3 the open loop touches the bounds at o = 20, so that no overdesign should be expected. This is shown in the simulation in Fig. 4, where the its maximum is about 8dB which is about the specification on d, at that frequency. This guides the designer in how to eliminate overdesign, i.e. by shaping the controller to be as close as possible to the bounds. 6. Conclusions The correlation between gain and phase margin and a system’s time response is well-known in SISO systems. In this paper, the definition of the SISO gain and phase margin is extended to the plant input of a MIMO system with non-diagonal controllers. The QFT design procedure for uncertain MIMO feedback systems has been extended in order to satisfy gain and phase margin performance (which can be a function of w) at the plant input. Thus the transfer function from each plant input into itself will have the features of a SISO system with the same gain and phase margins. Compact equations for bounds calculated at each design step are presented. References Astrom, K. J. (1985). Adaptive control-a way to deal with uncertainty. In J. Ackerman (Ed.), Uncertainty and Control Lecture Notes in Control and Information Science, p. 70, Springer, Berlin. Davison. E. J. (1986). Robust control for industrial svstems. In P&c. of 2h Ck, WPI, Athens, Greece. . Desoer, C. and M. Vidyasagar (1975). Feedback Systems: Input-Output Properties. Academic Press, New York. Francis, A. B. (1987). A course in H, control theory. In Lecture Notes in Control and Information Sciences. p. 88. Springer, Berlin.
Rosenbrock, H. H. (1974). Computer Aided Control System Design. Academic Press, London. Sobel, K. M., J. C. Chung and E. Y. Shapiro (1983). Application of MIMO phase and gain margins to the evaluation of a flight control system. In Proc. ACC. 3, 1286-1287. Tannenbaum, A. (1986). On the multivariable gain margin problem. Automatica, 22, 381-383. Yaniv. 0. (1991). Arbitrarily small sensitivity in multipleinput-output uncertain feedback systems. Automatica, 27, 565-568. Yaniv. 0. (1992). Synthesis of uncertain MIMO feedback systems for gain and phase margin at different channel breaking points. Automatica, 28, 1017-1020 Yaniv. 0. and I. Horowitz (1986). A quantitative design method for MIMO linear feedback systems having uncertain plants. ht. J. Contr., 43, 401-421. Youla, D. C. and J. J. Bongiorno (1985). A feedback theory of two degree-of-freedom optimal Wiener-Hopf design. IEEE Trans. Autom. Contr., AC-SO, 652-665. Appendix Proof of equafion (8). Trivial consequence p. 655), applied on the right side of (I+WGP]
of Kailath
‘=[l+W(GpG,)P)+WG,P]-’ = [C + g,uu’]
‘.
Proof of equation (10). Since rows and columns zero and G” is diagonal I + WG”P
(1980,
= I + W,“G”P,
= I + W,G,P,,
(A.1) (Yof G” are
(A.2)
where G, is the same as G, except for the diagonal zero elements, which are replaced by 1. Now use Kailath (1980, p. 656) and substitute G,’ = G, ‘.