(Vidyasagar, 1985) that this problem is equivalent to simultaneously stabilizing n plants with a stable compensator C(s), i.e. the simultaneous strong stabilization.
STRONG SIMULTANEOUS STABILIZATION OF n SISO PLANTS C. Abdallah, P. Dorato Department of EECE University of New Mexico Albuquerque, NM 87131, USA and M. Bredemann Sandia National Laboratories Div. 9222, Mail Stop 0972 Albuquerque, NM 87185, USA
ABSTRACT In this brief paper 1 we present sucient conditions for the existence of a single stable controller to stabilize a set of n plants: P1 ; P2 ; :::; Pn (strong simultaneous stabilization). As is well known this is equivalent to the existence of a single controller, not necessarily stable, to stabilize n + 1 plants (simultaneous stabilization). The basic assumption required in the current paper is that all the plants have the same unstable zeros. A practical example of where such an assumption holds, is when each plant has the same pure delay which is approximated by a non-minimum phase rational function. The results here are applied to examples of this type.
Key Words. Simultaneous stabilization, linear systems, interpolation algorithm.
1 This paper was rst presented at the IFAC Symposium on Robust Control Design, Rio De Janeiro, 14{16 September, 1994.
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1 INTRODUCTION The problem of stabilizing n + 1 di erent plants is a longstanding problem in the robust control literature. The problem is relevant in applications where the plant is only known to belong to a set of n +1 di erent plants, or where the failure of sensors or actuators will drastically change the plant from its current description. More recently, the problem has been studied in conjunction with the problem of stabilizing a nonlinear plant, which is linearized about n +1 operating points. It has been shown (Vidyasagar, 1985) that this problem is equivalent to simultaneously stabilizing n plants with a stable compensator C (s), i.e. the simultaneous strong stabilization of n plants. Unfortunately, this latter problem is yet unsolved except in the case where n = 1 (Vidyasagar, 1985). There exist necessary and sucient conditions and a synthesis procedure for solving the strong stabilization of one plant as described in (Youla et al., 1974), namely a plant is strongly stabilizable if and only if it satis es the Parity-Interlacing-Property (PIP). For the case, where n > 2 few results have appeared in the direction of a general solution to the simultaneous stabilization problem. In (Ghosh, 1986) for example, sucient conditions are found for the simultaneous stabilization of 3 di erent plants. In (Barmish and Wei, 1985), the case of n minimum-phase plants, which have the same sign in their high-frequency gains is treated. A similar problem was treated in (Chapellat and Bhattacharyya, 1988). More recently, e orts towards the general solution of the problem have resulted in the special results in (Wei, 1990). Recently and in the spirit of (Wei, 1990), Blondel et al. have presented necessary conditions for the simultaneous stabilizability of more than 2 plants (see, e.g., (Blondel et al., 1991)). Unfortunately, most of these conditions, even the so-called necessary and sucient ones (with the exception of (Barmish and Wei, 1985)and (Chapellat and Bhattacharyya, 1988)) are not computable. In other words, the conditions e ectively translate the problem into another unsolved problem. In fact, Blondel has now shown (Blondel, 1994) that the problem of simultaneously stabilizing more than two plants is undecidable by rational operations. Our only hope then is to enlarge the class of systems for which sucient conditions for simultaneous stabilizability exist. In this paper, we describe computable conditions for solving the strong simultaneous stabilization problem for a class of n plants. The results are constructive and a rational compensator may ultimately be computed. This paper is organized in the following manner. Section 2 de nes the problem and review the available results. The solution to the special strong simultaneous stabilization problem is given in section 3, and some numerical examples are presented in section 4. Finally, our conclusions are presented in section 5.
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2 PROBLEM STATEMENT The problem addressed in this paper is the following: Given n single-input-singleoutput (SISO) plants P1 (s); P2 (s); ; Pn (s), does there exist a single stable compensator C (s) such that the closed-loop (unity feedback) system is internally stable for any of the given plants. As is well known, see for example (Vidyasagar, 1985), the closed-loop systems are internally stable if and only if each of the three transfer functions 1 Pi (s) C (s) (1) 1 + Pi (s)C (s) ; 1 + Pi (s)C (s) ; 1 + Pi (s)C (s) are bounded-input-bounded-output (BIBO) stable. The strong simultaneous stabilizing compensator C (s) must then make all of the above transfer functions stable. Let us rst recall some of the properties and available results to be used in the sequel. Any SISO rational function P (s) may be written as the ratio of two stable, co-prime rational functions N (s)=D(s) (Vidyasagar, 1985). The set of all proper, Bounded-input-bounded-output (BIBO) stable rational functions is denoted by H 1. A Unit of H 1 is a rational function U (s) 2 H 1 whose inverse is also in H 1 , i.e. a Unit is a stable rational function whose inverse is also stable (Vidyasagar, 1985). Therefore, a Unit is necessarily exactly proper. We let k P (s) k1 denote the H 1 norm of functions in H 1, and recall by the maximum-modulus theorem (Vidyasagar, 1985) that
k P (s) k1 = supfj P (s) j ; Re(s) 0g = supfj P (jw) j ; w 2 Rg (2) Note nally, that a Unit U (s) will attain both its maximum and its minimum magnitude along the jw axis (Walsh, 1956).
Theorem 1 The Parity-Interlacing-Property (PIP) (Youla et al., 1974) A
linear system P (s) is stabilizable with a stable compensator C (s) or strongly stabilizable with C (s) if and only if the number of poles of P (s) between any pair of real zeros in the right-half-plane is even. Moreover, and for the case where all the unstable zeros of N (s) are simple, the set of all stable, stabilizing compensators is given by ) ? D(s) C (s) = U (sN (s)
where N (s); D(s) 2 H 1 , U (s) is a Unit of H 1 such that
U ( i ) = D( i) at all the right-half plane (RHP) zeros i of N (s) including those at 1.
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Theorem 2 (Vidyasagar, 1985) A compensator C (s) = DN
c (s) c (s)
where Nc (s); Dc(s) 2
H 1 internally stabilizes the plant P (s) = Np (s)=Dp(s) if and only if Nc(s)Np(s) + Dc(s)Dp(s) is a Unit of H 1 .
Theorem 3 (Vidyasagar, 1985) If C (s) is stable, then it internally stabilizes the plant N (s)=D(s) if and only if D(s) + C (s)N (s) is a Unit in H 1. In general, the problem of strong stabilization of 2 plants is not completely solved. In this paper, we present sucient conditions for strong simultaneous stabilization of a class of n plants. These conditions and constructive and a rational compensator may ultimately be computed.
3 STRONG SIMULTANEOUS STABILIZATION OF n PLANTS Consider the set of plants Pi (s); i = 1; 2; :::; n where Ni (s) Pi(s) = D (3) i (s) where Ni(s); Di (s) 2 H 1 . As reviewed in Theorem 1, P1 (s) is strongly stabilizable if and only if it satis es the PIP property and the stabilizing compensator is given by (4) C (s) = U1(sN) ?(sD) 1(s) 1 where U1 (s) is a Unit which interpolates to D1 at the RHP zeros of N1. Let
Ni = Ni+Ni? ; i = 1; 2; :::; n
(5)
where Ni+ contains all RHP zeros and Ni? contains all LHP zeros. We then state the following assumption.
Assumption A1: The class of plants considered has the following properties: 1. All Pi (s) have the same relative degree, 2. Ni+ (s) = Nj+ (s); 8i; j = 1; ; n. In other words, the plants have the same RHP zeros. 4
We now choose a plant P1 (s) with the common relative degree n , and form all Ni (s); Di(s). Let us de ne the following quantities to be used later ? ? Wi(s) = N1 (s)Di(s) ?? Ni (s)D1(s) ; i = 2; ; n (6) Ni (s) Note that Wi (s) in (6) is in H 1 since it is proper, and since Ni? (s) has no zeros in RHP. Then, let us de ne W (s) to be a unit in H 1 which satis es the inequalities jWi(j!)j jW (j!j; all ! (7) Also, let 1k denote the RHP zeros of N1(s), and W ( 1k )
k1 = D (8) 1 1 ( k ) We will assume in the sequel that all k1 are positive for all real positive 1k , since negative k1 may be simply accommodated by computing a unit V (s) that interpolates ? k1 , and letting U (s) = ?V (s). The main result of this short paper is given
in theorem 4 below. For the sake of simplicity we assume all the plant zeros in the RHP are nite and simple. The important case of zeros at in nity, with possible multiplicity greater than one, will be discussed separately in section 4. Theorem 4 The plants Pi(s) ; i = 1; 2; :::; n, each satisfying the PIP, are simultaneously strongly stabilizable if 1. All Pi (s) satisfy assumption A1, 2. The Hermitian matrix H with entries # " ln( k1) + ln( j1) (9) ? 1k + 1j
is positive de nite. Proof: Let us assume that our plants satisfy A1 and the PIP, and choose a plant P1 (s). Using Theorem 1, we nd a stable, stabilizing compensator C (s) = U1(sN) ?(sD) 1 (s) (10) 1
where U1 (s) interpolates to D1( 1k ) at the RHP zeros 1k ; k = 1; ; r1 of N1 (s). Let us then use the same compensator to stabilize Pi (s) for i 6= 1. By calling on Theorem 3, we require that Di (s) + U1 (sN) ?(sD) 1 (s) Ni (s) (11) 1 5
be a Unit Ui (s). Therefore,
Ui (s) = Di(s) + U1(sN) ?(sD) 1(s) Ni(s) 1 N N i (s) 1 (s)Di (s) ? Ni (s)D1 (s) = U1 (s) N (s + N1(s)U1(s) 1 N 1 (s)Di (s) ? Ni (s)D1 (s) = U1 (s) Vi(s) + Vi(s) N (s)U (s) i
1
# ? ? N 1 (s)Di (s) ? Ni (s)D1 (s) = U1 (s)Vi(s) 1 + N ? (s)U1(s) "
i
W i (s) = U1 (s)Vi(s) 1 + U (s) 1 where Vi (s) = Ni (s)=N1(s) is guaranteed to be a Unit since
(12)
Ni (s)=N1(s) = Ni?(s)=N1?(s)
(13)
Therefore, in order to make Ui (s) a Unit, it is necessary and sucient to nd U1(s) such that i (s) 1+ W (14) U1 (s) is a Unit, where Wi (s) is as de ned in (6). The original problem has then been restated as that of the simultaneous stabilization of a set of stable plants, Wi (s), by a Unit compensator. A solution to this problem is currently unavailable, so we turn instead to a sucient condition for the stabilizability of all plants by using a \small gain" theorem, i.e. by requiring
k Wi(s)=U (s) k1< 1 (15) We let V (s) = 1=U (s), also a Unit since U (s) is a Unit. Using the bounding properties of W (s), see (7), we can write k Wi(s)V (s) k1k W (s)V (s) k1 (16) 1
1
1
which then guarantees that (15) holds if we make,
k W (s)V (s) k1< 1; 8i = 1; ; n (17) The problem then reduces to nding a unit U (s) = W (s)V (s) which interpolates to k = W ( k )=D ( k ), and also has an H 1 norm less than one. Note that if the PIP property is not satis ed for each plant, the magnitude of Wi ( k )=U ( k ) will exceed unity, and there can exist no unit U (s) with H 1 norm bounded by one. 1
1
1
1
1
6
1
1
The problem of nding a bounded unit which interpolates to given points is solved in (Tannenbaum, 1982) so we will proceed in the same fashion. We choose U (s) = e?Z(s) ; (18) where Z (s) is a strictly positive real (SPR) function (Dorato et al., 1989). Since Z (s) has a positive real-part in the RHP , U (s) is a Unit whose H 1 norm is bounded by 1, and U (s) will interpolate to the desired points if and only if Z (s) interpolates to ?ln ( k1). Condition 2) in the theorem then becomes the necessary and sucient conditions (Youla and Saito, 1967) for the existence of Z (s) and thus of V (s). Once U (s) is found, we can obtain the desired U1 (s) = W (s)=U (s). Finally, U (s) may be approximated by a rational function, to insure a rational compensator, as discussed in (Tannenbaum, 1982), or a rational interpolating Unit may be found directly by using the algorithm in (Bredemann, 1995).
Note 1 The methodology of Theorem 4 may also be used to stabilize an in nite family of plants
P = fP (s; q) = N (s; q)=D(s; q); q 2 Qg assuming all members of the family satisfy assumption A1 and the PIP, and that condition 2) of Theorem 4 is satis ed. In other words,we assume that the unstable zeros of P (s; q ) do not change as q changes in Q and such that for some q1 2 Q, ? ? W (s; q) = N (s; q1)D(s;Nq)??(s;Nq) (s; q)D(s; q1) ; 8q 2 Q and a Unit W (s) which guarantees
jW (j!; q)j jW (j!j; all !; all q 2 Q exists.
4 NUMERICAL EXAMPLES We present next two examples to illustrate the applicability of our results.
Example 1.
Consider the problem of simultaneously stabilizing the two plants ? s)(2 ? s) ; P (s) = (1 ? s)(2 ? s) P1(s) = (1 (1 2 ? 9s)2 81s2 with a stable compensator. Let 2 2 D1(s) = (1(1?+9ss))2 ; D2(s) = (181+ss)2 7
18s?1 and a Unit which bounds W2 (s) is given by Then W2(s) = (1+ s)2
:01s + 1) W (s) = (18s +(s1)(0 + 1)2 so that
W (1) = 0:2998; 1 = W (2) = 0:1306
11 = D 2 (1) D (2) 1
1
The Hermitian matrix in (9) is then given by "
1:2045 1:0801 1:0801 1:0179
#
Since this matrix is positive de nite, a solution to the strong simultaneous stabilization problem exists for the two plants given above. Using the interpolation theory for positive real functions in (Youla and Saito, 1967) one obtains the SPR function (1:7695s + 0:2305) Z (s) = 1:2045 (0 :2305s + 1:7695) which interpolates to the points
Z (1) = ?ln( 11) = 1:2045; Z (2) = ?ln( 21) = 2:0356 as required. The algorithm described in (Bredemann, 1995) directly yields the rational interpolating Unit, 10?6 (s + 53:66)5 U (s) = 1:80628 (s + 3:94)5 and the stable fth-order compensator s2 + 11:8797s + 85:011) C (s) = 99; 571 (s + 100)(s + 10:747)(s (+s 0+:00444)( 53:66)5
Example 2.
Consider the problem of designing a stable compensator which stabilizes a pendulum, with delayed control e ort, about three positions, = 0; =2; and . The nonlinear dynamics of the pendulum are given by + l sin = 1 T (t ? )
g
l2m
where is the angular position of the pendulum, T is the input torque, and is the control-e ort time-delay. If the pure time-delay term e? s is approximated by 8
the rst-order rational function (1 ? s=2 )=(1 + s=2 ) and the pendulum dynamics are linearized about the three points above, one obtains with the parameter values l=g = 1 and = 1, the three plant transfer functions ?s P1 (s) = (s2 ?21)(2 + s) 2 ? s P2(s) = s2 (s + 2) ?s P3(s) = (s2 +21)(2 + s) where P1 (s); P2(s) and P3(s) correspond to the linearizations about = ; =2, and 0 respectively. If we let N1(s) = N2 (s) = N3(s) = h2 ?(ss) 1 2 2 2 D1(s) = sh(?s)1 ; D2 (s) = hs(s) ; D3(s) = sh(+s)1 where h1 (s) = (s + 2)3 and h(s) = (s + 2)2, then for a stable compensator we must satisfy the interpolation conditions U1 (2) = D1 (2) = 3=16 U1(1) = D1(1) = 1 U10 (1) = D10 (1) = ?4 (19) where the notation U10 (s) is used to indicate the derivative of U1 (s) with respect to the variable s?1 . The last two conditions in (19) are required to insure that the term U1 (s) ? D1(s) has relative degree equal to 2 at in nity. A Unit W (s) which bounds Wi (s) in this case is given by s + 1)2 W (s) = 2 (0(:01 s + 2)2 Using
W (s) = U1 (s)e?Z(s) W 0 (s) = U1 (s)e?Z(s) Z 0(s) + e?Z(s) U10 (s) the interpolation conditions on U1 (s) are transferred to the following interpolation conditions on Z (s): Z (2) = 0:3662; Z (1) = 8:517; Z 0 (1) = 200 from the interpolation theory for positive real functions given in reference (Youla and Saito, 1967), appropriately extended to interpolation points at in nity, one 9
may verify that a solution exists and that the required interpolating SPR function is given by
:516s2 + 3; 302:64s + 4 Z (s) = 0:3662 46 2s2 + 95:04s + 6; 597:28 From this Z (s) one can compute U1(s), and nally from (10) and (18) an irrational compensator C (s). However, from the algorithm in (Bredemann, 1995), the following rational interpolating Unit is obtained 55851)(s2 + 444:362s + 6:25 1010) U (s) = 0:0002 ((ss++14 :105)(s2 + 500; 000s + 6:25 1010) which results in the stable third-order compensator (s + 1:7695)(s + 2)(s + 1:4072 105) C (s) = 2:4711 1010 (s + 55; 851)(s2 + 444; 362s + 6:25 1010)
5 CONCLUSIONS In this paper, we have characterized a class of SISO systems which are strongly simultaneously stabilizable. Our conditions are checkable and when satis ed, a compensator may be found to achieve the stabilizability goal. Note that stabilizability with respect to other regions (Unit circle, or any closed complex region), can be derived in a similar fashion.
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Dorato, P., Park, H., and Li, Y. (1989). An algorithm for interpolation with units in H 1 with applications to feedback stabilization. Automatica, 25:427{430. Ghosh, B. (1986). Simultaneous partial pole-placement: A new approach to multimode design. IEEE Trans Auto. Control, AC-31(5):440{443. Tannenbaum, A. (1982). Modi ed Nevalinna-Pick interpolation and feedback stabilization of linear plants with uncertainty in the gain factor. Int. Jour. Control, 36(2):331{336. Vidyasagar, M. (1985). Control System Synthesis: A Factorization Approach. The MIT Press, Cambridge, MA, 1st edition. Walsh, J. (1956). Interpolation and Approximation by Rational Functions in the Complex Domain. American Mathematical Society, Rhode Island, NY, 2nd edition. Wei, K. (1990). Stabilization of a linear plant via a stable compensator having no real unstable zeros. Systems and Control Letters, 15(3):259{264. Youla, D., Bongiorno, J., and Lu, C. (1974). Single-loop feedback stabilization of linear multivariable dynamical systems. Automatica, 10:159{173. Youla, D. and Saito, M. (1967). Interpolation with positive real functions. J. Franklin Inst., 284:77{108.
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