- Email: [email protected]
Pole assignment by output feedback
Automatica, Vol. 29, No. 6, pp. 1599-1601, 1993
001)5-1098193 $6,00 + 0.00 (~ 1993 Pergamon Press Ltd
Printed in Great Britain.
Technical Communique
Pole Assignment by Output Feedback: A Solution for 2 × 2 Plants* QING-GUO W A N G , t TONG HENG LEEr and C H A N G C H I E H H A N G t Key Words--Pole assignment; output feedback; muitivariable systems; polynomial approach; controller parametrization.
performance specifications such as the sensitivity reduction to model uncertainties and disturbances. An example of order 4 is provided for illustration and it violates the condition, m + l > n. However, with our method, it is still possible to assign all four closed-loop poles exactly.
Abstract--The problem of pole assignment by static output feedback is solved in this paper for 2 x 2 plants. It is shown that the problem solvability is equivalent to the solvability of some quadratic equation and the latter can be checked through the affine transformation of quadratic functions. All solutions are given in a parametric form if a solution exists. An example is provided to illustrate the method.
2. Problem formulation and solution Consider a two-input, two-output system of order n described by
1. Introduction THE PROBLEM OF pole assignment by output feedback has been one of long standing interest and, unlike the less difficult problem of state feedback, has never really been completely solved. Brasch and Pearson (1970) have shown that it is possible to place all closed-loop poles arbitrarily using a dynamic compensator of order r = rain {re - 1, ro 1}, where rc and ro are the system controllability and observability indices, respectively. A frequency-domain approach to the problem has been described by Seraji (1980). It follows from the seminal work of Kimura (1975) and development by Fletcher (1980) that the problem by dynamic compensator can be reduced to one by static feedback for the augmented system in which the compensator has been adjoined to the original plant and the inequality condition, m + l > n, is crucial for the complete assignment of n distinct eigenvalues to a controllableobservable/-input, m-output system of order n. In this case, a parameterization of all such feedback controllers has been developed by Roppenecker and O'Reilly (1989). However, when m + l <- n, the (n - m - l + 1) poles cannot be assigned and they may move to undesirable locations. The algorithm by Chen et al. (1988) can be used to assign (max {m, l) - 1) poles arbitrarily and find the best locations for the remaining unassigned poles with the help of the root locus method. In Ramar and Appukuttan (1991), another procedure has been given to design a constant output feedback matrix which assigns (m + l - 2) poles exactly and shifts all the unassigned poles to suitable locations using root locus techniques. In this paper, we consider the problem of pole assignment by static output feedback for 2 x 2 plants. The feedback matrices to be used are not restricted to unity-rank but are generally of full rank. The closed-loop systems are allowed to have multiple eigenvalues. Both cases, m + l > n and m + I-< n, are treated in a unified framework. It is shown that the problem solvability is equivalent to the solvability of some quadratic equation in several variables and the latter can be checked through the affine transformation of quadratic functions. Moreover, if the problem is solvable for a given plant, all solutions are parametrized in terms of the solutions to the quadratic equation so that these remaining free parameters can further be used to meet other
y(s) = C(s)u(s),
(1)
where G(s) is a strictly proper rational transfer function matrix. A static output feedback u(s) : - K y ( s )
(2)
is applied to the system. It is well known (Rosenbrock, 1970) that the closed-loop characteristic polynomial pc(s) can be expressed as
pc(s) = po(s) det [I + G(s)K], where po(s) is the monic n-degree characteristic polynomial of the system G(s). The pole assignment problem to be solved in this paper is to determine whether or not there exists a constant K such that the closed-loop system has the desired monic n-degree polynomial pd(S) as its characteristic polynomial; that is, pd(S) =po(s) det [1 + G(s)K].
(3)
It follows from matrix theory that Po det [1 + GK] =Po det [M + GK] Ix=t =po[A 2 + trace (GK)~, + det (GK)] Ix=t =po[1 + trace (GK) + det (GK)]. Write G and K element-wise as
then (3) becomes fii(s)ki = h(s)
(4)
i=l
where f A s ) := p o(S )g,(s )
=:A,s"-t + f ~ s " -
2+ . . .
+f.,. i = 1, 2, 3, 4
A(s) := po(s) det (G(s)) =:flss " - ' +f25s n-2 + - . . +fn5 k 5 := det (K) = klk 4 - k2k 3
h(s) := pd(s) -- po(s) * Received 8 December 1992; recommended for publication by Editor W. S. Levine. t Department of Electrical Engineering, National University of Singapore, Singapore 0511, Republic of Singapore.
= : hi s n - I + h2 $n-2 + ' " • +
h~.
Equating the coefficient of the same powers on both sides of (4) yields Fk = h 1599
(5)
1600
Technical
This agrees with the results of Kimura (1975) and Roppenecker and O'Reilly (1989). The exception is that one of the very restrictive conditions, (i) or (ii) in Theorem 1, is satisfied. This possibility has also been identified by Kimura (1975). In view of the above observations, the most critical case is n = 4 and if r = n = 4 , (8) is then reduced into the following form
where
[A, f,~ F=~!
k=
I
f22
' ""
f.-] f25
n2
" ' ":
fLJ'
[i'] [i1] k2
,
h=
h2
5
ak~ + b k 5 + c =0.
.
It has a real solution if and only if
n
b 2 - - 4ac >- O,
Note that (5) has the constraint (6)
k 5 = k l k 4 - k e k 3.
As a result, our pole assignment problem is solvable if and only if equations (5) and (6) have a common solution for k. We now have to determine the simultaneous solvability of (5) and (6). For this, it is obviously necessary that (5) alone has a solution. It implies that Rank (F) = Rank [F h] = : r.
[Fll F = [F21
kl = Fill(hi lh = [hi h z " "
Consider the plant in Ramar and Appukuttan (1991) described by the state space model 5( = A X + B U
Ftz]
F22J
Y= CX
with F~1 being of size r x r and nonsingular. The solutions to (5) are then given by
k I = [kl
which is very easy to check. From the point of view of application, controller design can be carried out as follows. The solvability condition in Theorem 1 is first checked and a solution z is computed if it is solvable. It is then substituted into (9) to get k2 and the remaining parameters are obtained from (7). The procedure also shows that all solutions to the given pole assignment problem are parametrized in terms of solutions to (10) or (11). 3. A n example
If this is the case, we can assume, without loss of generality, that F has the partition
where
Communique
-
F12k2),
with
(7) A=
k 2 = [k,+l k,+2" ' " ks] r, and h,] r. By the substitution of (7) into (6), (6) k 2 ' " '
k , ] T,
has the form
0
0
0
0
' 0
and kTAk2
+
brk2 + c = 0,
where A, b, and c are computed from F and h. The left side of (8) is a real quadratic function in (5 - r) variables and it follows from Birkhoff and MacLane (1977) that there exists an affine transformation k2 =
Pz + t
Its transfer function matrix is 1 I-s3+s 2 s + l ]
G(s)=~[ s3
(9)
with P nonsingular such that (8) is reduced to one of the following two forms: (z,)2 + • + ( ~ v ) 2 - (z~+,) 2 . . . . .
0 , 0 0]
(8)
(zo+.,y + z . . . . . , = 0 ,
(10)
and the open-loop characteristic polynomial is po(s) = det (sl - A ) = s 4.
It follows from definition that
or
fl(S)
(ZI) 2 + " + ( Z v ) 2 -- (Zv + 1) 2 . . . . .
(Zv + s ) 2 + d = 0,
sZ+s ],
= S 3 + S 2,
(11) f 2 ( S ) : S + 1,
where d is constant. We can thus state our solvability condition as follows.
A(s) = s 3, A(s) = s ~ + s,
Theorem 1. For a 2 × 2 plant, the pole assignment problem
by static output feedback is unsolvable if and only if either (5) has no solution, or the afline transformation (9) reduces (8) to the form given in (11) with one of the following conditions, (i) or (ii), holding true (i) v = 0 a n d d < 0 ; (ii) s = 0 and d > 0. It can be noted that because k has dimension equal to 5 the quadratic function on the left side of (8) comprises very few variables so that the computations required in Theorem 1 for solvability checking should not be very involved. The development above also provides us with intuition on the solvability of any problem involving pole assignment by output feedback. One knows that the problem is normally unsolvable when n > 4 because the number of free parameters in K is 4, less than the number n of poles to be assigned. In this case, one sees that (5) generally has no solution if n > 5, or possibly has a unique solution if n = 5, since h is specified by the designer arbitrarily. However, this unique solution rarely satisfies the additional equation (6). If n < 4, the left side of (8) is a quadratic function in more than two variables and the equation is likely to have solutions.
A(s) =
s + 1.
Ramar and Appukuttan (1991) assigned two closed-loop poles at - 2 and - 3 . Assume first that we want to place the additional two poles at - 4 and - 5 . It then follows that pu(s) = (s + 2)(s + 3)(s + 4)(s + 5)
= s 4 + 14s 3 + 71s 2 + 154s + 120 and h(s) = 14s 3 + 71s 2 + 154s + 120.
For this example, (5) becomes
[i
00 01 01 0t 1 0 1 1
0
0
I)rl k2
kk~ =
14 71 154 " L120/
k5
One sees that F H consisting of the first four columns of F is
Technical Communique nonsingular and the solutions to the above equation are given by [k~l [ 0] [ 37 1 k2 = - 1 120 k3 0 k5 + -23 " k4 L 0_1 L 34_J Substituting it into (6) yields 24k5 - (37 x 34 + 23 x 120) = 0 and it has the real solution ks = 167.4167. The required feedback gain matrix is then computed as 37 K = [-47.4167
-23 34]'
which is nonsingular. One can verify that the closed-loop matrix, ( A - BKC), indeed has the desired eigenvalue set {-2, - 3 , - 4 , -5}. We now change the desired closed-loop pole set into { - 2 , - 3 , - 5 , - 5 } , which means that the system will have a multiple pole at - 5 . The feedback matrix which assigns these poles to the system still exists and it is 46 K = [-45.625
-31 35]'
However, for some particular sets of poles, equation (6) may not have a solution. To see this, consider next the case pd(S) = (S + 2)(S + 3)(S + 4)(S + q),
q > O.
Through simple calculations, equation (6) for the present case is given by 6(q - 1)k s = 158q 2 + 4q + 48. It has no solution for q = 1 but otherwise it is always solvable. This means that the closed-loop pole set,
1601
{-1,-2,-3,-4}, cannot be assigned exactly however arbitrarily close, a case which was shown by Kimura (1975) to happen if m + l > n. 4. Conclusions In this paper, the problem of pole assignment by static output feedback has been solved for 2 x 2 plants. A necessary and sufficient condition for solvability of the problem is established and all solutions to the problem are parametrized in terms of the solutions to a quadratic equation. The computations involved are simple and possible to implement on microcomputes. From a theoretical point of view, two-input two-output plants are a small class of general multivariable systems, but they are commonly encountered in industry and the results presented here will thus be quite useful for practical applications. References Birkhoff, G. and S. MacLane (1977). A Survey of Modern Algebra, 4th ed. Macmillan, New York. Brasch, F. M. and J. B. Pearson (1970). Pole placement using dynamic compensators. IEEE Trans. Aut. Control, AC-15, 34-43. Chen, C.-L., T. C. Yang and N. Munro (1988). Output feedback pole-assignment procedure, lnr J. Control, 48, 1503-1518. Fletcher, L. g. (1980). An intermediate algorithm for pole-placement by output feedback in linear multivariable control systems. Int. J. Control, 31, 1121-1136. Kimura, H. (1975). Pole assignment by gain output feedback. IEEE Trans. Aut. Control, AC-20, 509-516. Ramar, K. and K. K. Appukuttan (1991). Pole assignment for multi-input multi-output systems using output feedback. Automatica, 27, 1061-1062. Roppenecker, G. and J. O'Reilly (1989). Parametric output feedback controller design. Automatica, 25, 259-265. Rosenbrock, H. H. (1970). State Space and Multivariable Theory. Nelson, London. Seraji, H. (1980). Design of pole-placement compensators for multivariable systems. Automatica, 16, 335-338.
001)5-1098193 $6,00 + 0.00 (~ 1993 Pergamon Press Ltd
Printed in Great Britain.
Technical Communique
Pole Assignment by Output Feedback: A Solution for 2 × 2 Plants* QING-GUO W A N G , t TONG HENG LEEr and C H A N G C H I E H H A N G t Key Words--Pole assignment; output feedback; muitivariable systems; polynomial approach; controller parametrization.
performance specifications such as the sensitivity reduction to model uncertainties and disturbances. An example of order 4 is provided for illustration and it violates the condition, m + l > n. However, with our method, it is still possible to assign all four closed-loop poles exactly.
Abstract--The problem of pole assignment by static output feedback is solved in this paper for 2 x 2 plants. It is shown that the problem solvability is equivalent to the solvability of some quadratic equation and the latter can be checked through the affine transformation of quadratic functions. All solutions are given in a parametric form if a solution exists. An example is provided to illustrate the method.
2. Problem formulation and solution Consider a two-input, two-output system of order n described by
1. Introduction THE PROBLEM OF pole assignment by output feedback has been one of long standing interest and, unlike the less difficult problem of state feedback, has never really been completely solved. Brasch and Pearson (1970) have shown that it is possible to place all closed-loop poles arbitrarily using a dynamic compensator of order r = rain {re - 1, ro 1}, where rc and ro are the system controllability and observability indices, respectively. A frequency-domain approach to the problem has been described by Seraji (1980). It follows from the seminal work of Kimura (1975) and development by Fletcher (1980) that the problem by dynamic compensator can be reduced to one by static feedback for the augmented system in which the compensator has been adjoined to the original plant and the inequality condition, m + l > n, is crucial for the complete assignment of n distinct eigenvalues to a controllableobservable/-input, m-output system of order n. In this case, a parameterization of all such feedback controllers has been developed by Roppenecker and O'Reilly (1989). However, when m + l <- n, the (n - m - l + 1) poles cannot be assigned and they may move to undesirable locations. The algorithm by Chen et al. (1988) can be used to assign (max {m, l) - 1) poles arbitrarily and find the best locations for the remaining unassigned poles with the help of the root locus method. In Ramar and Appukuttan (1991), another procedure has been given to design a constant output feedback matrix which assigns (m + l - 2) poles exactly and shifts all the unassigned poles to suitable locations using root locus techniques. In this paper, we consider the problem of pole assignment by static output feedback for 2 x 2 plants. The feedback matrices to be used are not restricted to unity-rank but are generally of full rank. The closed-loop systems are allowed to have multiple eigenvalues. Both cases, m + l > n and m + I-< n, are treated in a unified framework. It is shown that the problem solvability is equivalent to the solvability of some quadratic equation in several variables and the latter can be checked through the affine transformation of quadratic functions. Moreover, if the problem is solvable for a given plant, all solutions are parametrized in terms of the solutions to the quadratic equation so that these remaining free parameters can further be used to meet other
y(s) = C(s)u(s),
(1)
where G(s) is a strictly proper rational transfer function matrix. A static output feedback u(s) : - K y ( s )
(2)
is applied to the system. It is well known (Rosenbrock, 1970) that the closed-loop characteristic polynomial pc(s) can be expressed as
pc(s) = po(s) det [I + G(s)K], where po(s) is the monic n-degree characteristic polynomial of the system G(s). The pole assignment problem to be solved in this paper is to determine whether or not there exists a constant K such that the closed-loop system has the desired monic n-degree polynomial pd(S) as its characteristic polynomial; that is, pd(S) =po(s) det [1 + G(s)K].
(3)
It follows from matrix theory that Po det [1 + GK] =Po det [M + GK] Ix=t =po[A 2 + trace (GK)~, + det (GK)] Ix=t =po[1 + trace (GK) + det (GK)]. Write G and K element-wise as
then (3) becomes fii(s)ki = h(s)
(4)
i=l
where f A s ) := p o(S )g,(s )
=:A,s"-t + f ~ s " -
2+ . . .
+f.,. i = 1, 2, 3, 4
A(s) := po(s) det (G(s)) =:flss " - ' +f25s n-2 + - . . +fn5 k 5 := det (K) = klk 4 - k2k 3
h(s) := pd(s) -- po(s) * Received 8 December 1992; recommended for publication by Editor W. S. Levine. t Department of Electrical Engineering, National University of Singapore, Singapore 0511, Republic of Singapore.
= : hi s n - I + h2 $n-2 + ' " • +
h~.
Equating the coefficient of the same powers on both sides of (4) yields Fk = h 1599
(5)
1600
Technical
This agrees with the results of Kimura (1975) and Roppenecker and O'Reilly (1989). The exception is that one of the very restrictive conditions, (i) or (ii) in Theorem 1, is satisfied. This possibility has also been identified by Kimura (1975). In view of the above observations, the most critical case is n = 4 and if r = n = 4 , (8) is then reduced into the following form
where
[A, f,~ F=~!
k=
I
f22
' ""
f.-] f25
n2
" ' ":
fLJ'
[i'] [i1] k2
,
h=
h2
5
ak~ + b k 5 + c =0.
.
It has a real solution if and only if
n
b 2 - - 4ac >- O,
Note that (5) has the constraint (6)
k 5 = k l k 4 - k e k 3.
As a result, our pole assignment problem is solvable if and only if equations (5) and (6) have a common solution for k. We now have to determine the simultaneous solvability of (5) and (6). For this, it is obviously necessary that (5) alone has a solution. It implies that Rank (F) = Rank [F h] = : r.
[Fll F = [F21
kl = Fill(hi lh = [hi h z " "
Consider the plant in Ramar and Appukuttan (1991) described by the state space model 5( = A X + B U
Ftz]
F22J
Y= CX
with F~1 being of size r x r and nonsingular. The solutions to (5) are then given by
k I = [kl
which is very easy to check. From the point of view of application, controller design can be carried out as follows. The solvability condition in Theorem 1 is first checked and a solution z is computed if it is solvable. It is then substituted into (9) to get k2 and the remaining parameters are obtained from (7). The procedure also shows that all solutions to the given pole assignment problem are parametrized in terms of solutions to (10) or (11). 3. A n example
If this is the case, we can assume, without loss of generality, that F has the partition
where
Communique
-
F12k2),
with
(7) A=
k 2 = [k,+l k,+2" ' " ks] r, and h,] r. By the substitution of (7) into (6), (6) k 2 ' " '
k , ] T,
has the form
0
0
0
0
' 0
and kTAk2
+
brk2 + c = 0,
where A, b, and c are computed from F and h. The left side of (8) is a real quadratic function in (5 - r) variables and it follows from Birkhoff and MacLane (1977) that there exists an affine transformation k2 =
Pz + t
Its transfer function matrix is 1 I-s3+s 2 s + l ]
G(s)=~[ s3
(9)
with P nonsingular such that (8) is reduced to one of the following two forms: (z,)2 + • + ( ~ v ) 2 - (z~+,) 2 . . . . .
0 , 0 0]
(8)
(zo+.,y + z . . . . . , = 0 ,
(10)
and the open-loop characteristic polynomial is po(s) = det (sl - A ) = s 4.
It follows from definition that
or
fl(S)
(ZI) 2 + " + ( Z v ) 2 -- (Zv + 1) 2 . . . . .
(Zv + s ) 2 + d = 0,
sZ+s ],
= S 3 + S 2,
(11) f 2 ( S ) : S + 1,
where d is constant. We can thus state our solvability condition as follows.
A(s) = s 3, A(s) = s ~ + s,
Theorem 1. For a 2 × 2 plant, the pole assignment problem
by static output feedback is unsolvable if and only if either (5) has no solution, or the afline transformation (9) reduces (8) to the form given in (11) with one of the following conditions, (i) or (ii), holding true (i) v = 0 a n d d < 0 ; (ii) s = 0 and d > 0. It can be noted that because k has dimension equal to 5 the quadratic function on the left side of (8) comprises very few variables so that the computations required in Theorem 1 for solvability checking should not be very involved. The development above also provides us with intuition on the solvability of any problem involving pole assignment by output feedback. One knows that the problem is normally unsolvable when n > 4 because the number of free parameters in K is 4, less than the number n of poles to be assigned. In this case, one sees that (5) generally has no solution if n > 5, or possibly has a unique solution if n = 5, since h is specified by the designer arbitrarily. However, this unique solution rarely satisfies the additional equation (6). If n < 4, the left side of (8) is a quadratic function in more than two variables and the equation is likely to have solutions.
A(s) =
s + 1.
Ramar and Appukuttan (1991) assigned two closed-loop poles at - 2 and - 3 . Assume first that we want to place the additional two poles at - 4 and - 5 . It then follows that pu(s) = (s + 2)(s + 3)(s + 4)(s + 5)
= s 4 + 14s 3 + 71s 2 + 154s + 120 and h(s) = 14s 3 + 71s 2 + 154s + 120.
For this example, (5) becomes
[i
00 01 01 0t 1 0 1 1
0
0
I)rl k2
kk~ =
14 71 154 " L120/
k5
One sees that F H consisting of the first four columns of F is
Technical Communique nonsingular and the solutions to the above equation are given by [k~l [ 0] [ 37 1 k2 = - 1 120 k3 0 k5 + -23 " k4 L 0_1 L 34_J Substituting it into (6) yields 24k5 - (37 x 34 + 23 x 120) = 0 and it has the real solution ks = 167.4167. The required feedback gain matrix is then computed as 37 K = [-47.4167
-23 34]'
which is nonsingular. One can verify that the closed-loop matrix, ( A - BKC), indeed has the desired eigenvalue set {-2, - 3 , - 4 , -5}. We now change the desired closed-loop pole set into { - 2 , - 3 , - 5 , - 5 } , which means that the system will have a multiple pole at - 5 . The feedback matrix which assigns these poles to the system still exists and it is 46 K = [-45.625
-31 35]'
However, for some particular sets of poles, equation (6) may not have a solution. To see this, consider next the case pd(S) = (S + 2)(S + 3)(S + 4)(S + q),
q > O.
Through simple calculations, equation (6) for the present case is given by 6(q - 1)k s = 158q 2 + 4q + 48. It has no solution for q = 1 but otherwise it is always solvable. This means that the closed-loop pole set,
1601
{-1,-2,-3,-4}, cannot be assigned exactly however arbitrarily close, a case which was shown by Kimura (1975) to happen if m + l > n. 4. Conclusions In this paper, the problem of pole assignment by static output feedback has been solved for 2 x 2 plants. A necessary and sufficient condition for solvability of the problem is established and all solutions to the problem are parametrized in terms of the solutions to a quadratic equation. The computations involved are simple and possible to implement on microcomputes. From a theoretical point of view, two-input two-output plants are a small class of general multivariable systems, but they are commonly encountered in industry and the results presented here will thus be quite useful for practical applications. References Birkhoff, G. and S. MacLane (1977). A Survey of Modern Algebra, 4th ed. Macmillan, New York. Brasch, F. M. and J. B. Pearson (1970). Pole placement using dynamic compensators. IEEE Trans. Aut. Control, AC-15, 34-43. Chen, C.-L., T. C. Yang and N. Munro (1988). Output feedback pole-assignment procedure, lnr J. Control, 48, 1503-1518. Fletcher, L. g. (1980). An intermediate algorithm for pole-placement by output feedback in linear multivariable control systems. Int. J. Control, 31, 1121-1136. Kimura, H. (1975). Pole assignment by gain output feedback. IEEE Trans. Aut. Control, AC-20, 509-516. Ramar, K. and K. K. Appukuttan (1991). Pole assignment for multi-input multi-output systems using output feedback. Automatica, 27, 1061-1062. Roppenecker, G. and J. O'Reilly (1989). Parametric output feedback controller design. Automatica, 25, 259-265. Rosenbrock, H. H. (1970). State Space and Multivariable Theory. Nelson, London. Seraji, H. (1980). Design of pole-placement compensators for multivariable systems. Automatica, 16, 335-338.