On Synthesis of Practically Valid Gain-Scheduled Controllers

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(1)On Synthesis of Practically Valid Gain-Scheduled Controllers. Anders Helmersson Division of Automatic Control Department of Electrical Engineering Linköpings universitet, SE-581 83 Linköping, Sweden WWW: http://www.control.isy.liu.se E-mail: andersh@isy.liu.se 25th June 2003. OMATIC CONTROL AU T. CO MMU. NICATION SYSTE. MS. LINKÖPING. Report no.: LiTH-ISY-R-2528 Submitted to IEEE Transactions on Automatic Control Technical reports from the Control & Communication group in Link¨ oping are available at http://www.control.isy.liu.se/publications..

(2) Abstract Sufficient conditions for the existence of a gain scheduled controller for LPV systems is presented. It is shown that it is always possible to find a controller that depends on the parameters but which is independent of the rate of parameter variation. The conditions are three LMIs, which are similar to the conditions used for standard H∞ synthesis problems. In the general case, the controller needs to have twice as many states as the original system. If a lower-order controllers is searched for, a rateindependent controller still resists a convex formulation.. Keywords: Gain-scheduling control, linear matrix inequalities (LMI), linear parameter-varying (LPV) systems, robust control..

(3) 1. On Synthesis of Practically Valid Gain-Scheduled Controllers Anders Helmersson Department of Electrical Engineering Linköping University SE-581 83 Linköping, Sweden www: http://www.control.isy.liu.se email: andersh@isy.liu.se. Abstract Sufficient conditions for the existence of a gain scheduled controller for LPV systems is presented. It is shown that it is always possible to find a controller that depends on the parameters but which is independent of the rate of parameter variation. The conditions are three LMIs, which are similar to the conditions used for standard H∞ synthesis problems. In the general case, the controller needs to have twice as many states as the original system. If a lower-order controllers is searched for, a rate-independent controller still resists a convex formulation.. Gain-scheduling control, linear matrix inequalities (LMIs), linear parameter varying (LPV) systems, robust control, uncertain systems. I. I NTRODUCTION The gain-scheduling problem has been subject to research interest over recent years. The focus of this paper is on linear paramter-varying (LPV) controllers for systems with parametric dependencies. We assume that the parameters, θ, are bounded and with bounded rate of variation. The synthesis problem can be solved using three linear matrix inequalities (LMIs) [2], [6], [1]. These include matrix functions X and Y that both depend on θ. In addition they include their ˙ time derivatives, X˙ and Y˙ , which consequently depend on both θ and θ.. June 16, 2003. DRAFT.

(4) 2. One problem with the synthesis algorithms is that the controller may depend on the derivative of the parameters. This dependency is in general prohibitive since it requires real-time measure˙ In [1] a gain-scheduled controller that is independent of θ˙ is ments not only of θ, but also of θ. called practically valid. ˙ + N˙ M T where N and M are solutions The dependency on θ˙ stems from a nonzero term XY to XY + N M T = I. In [1] different options are discussed on how to satisfy these conditions. Essentially it boils down to either assuming that θ˙ = 0 or forcing X or Y to constant matrices. In previous work [1], [6], N and M are assumed to be square and nonsingular. The contribution of this paper is to show that the conditions on N and M apply generally without any specific h i assumption on the number of columns in N and M . Specifically, we can choose N = X I h i ˙ + N˙ M T = 0. Thus using and M = −Y I , which always satisfy XY + N M T = I and XY ˙ an LMI formulation we can always find a practically valid controller, which is independent on θ. The price we have to pay is that the controller has twice as many states as the original system. This paper is organized as follows. A new extended version of the elimination lemma is presented in Section II. This new result is then applied to the LPV synthesis problem in Section III. Finally, in Section IV the conclusions are presented. The notation used in the paper is fairly conventional. For real symmetric matrices Q, Q > 0 stands for positive definite and means that xT Qx > 0 for any x 6= 0. Similarly, Q < 0 means negative definite and Q ≥ 0 positive semidefinite (xT Qx ≥ 0 for any x). For any matrix A, AT denotes its transpose and A⊥ any matrix with the following properties: null AT = range A⊥ and A⊥ A⊥T > 0, where null and range denote the null space and range space of a matrix. Note that A⊥ A = 0 and that A⊥ exists if and only if A has linearly dependent rows. II. A N EW S OLVABILITY C ONDITIONS The standard elimination lemma [4], also called projection lemma, gives necessary and sufficient conditions for the existence of K such that Q + U KV T + V K T U T < 0 holds. The new lemma introduced here gives an additional sufficient condition for finding a common K if Q is chosen from a set Q while U and V are fixed. The proof of the lemma is analogous to the proof in [3, Lemma 7.2]. Lemma 1: Let Q ⊂ Rn×n be a closed set of symmetric matrices and let U ∈ Rn×m and. June 16, 2003. DRAFT.

(5) 3. V ∈ Rn×p be given. Then Q + U KV T + V K T U T < 0,. ∀Q ∈ Q. (1). has a common solution K ∈ Rm×p if (i) V ⊥ QV ⊥T < 0, (ii) U ⊥ QU ⊥T < 0, (iii) V ⊥ QU ⊥T = constant, hold for all Q ∈ Q. If V ⊥ or U ⊥ does not exist the corresponding conditions disappear and are assumed to be satisfied. Proof: We show the lemma by construction. Make a congruence transformation of (1) using i h a nonsingular post-multiplier T = T1 T2 T3 T4 where the transformed rows (and columns) correspond to the following disjunct spaces (1) range T1 = null U T ∩ null V T (2) range T2 = range U ∩ null V T (3) range T3 = null U T ∩ range V (4) range T4 = range U ∩ range V Then, (1), is equivalent to  ˜ Q  11  ˜T  Q12   ˜T  Q13   ˜T Q 14 where. ˜ 12 Q ˜ 22 Q ˜T + K ˜T Q 23 23. . ˜ 13 Q. ˜ 14 Q. ˜ 23 + K ˜ 23 Q ˜ 33 Q. ˜ 24 + K ˜ 24 Q ˜ 34 + K ˜T Q 43. ˜T ˜ ˜ T ˜T ˜ ˜ ˜T Q 24 + K24 Q34 + K43 Q44 + K44 + K44.     <0   . (2). .  h iT h i ˜ ˜ K K  23 24  = T2 T4 U KV T T3 T4 ˜ 43 K ˜ 44 K. (3). can be chosen freely. We first consider the the sub-matrix in (2) that comprises the first three rows and columns. ˜ 11 < 0 and Eliminating the first row and column by Schur complement yields Q   T ˜ −1 ˜ T ˜ −1 ˜ ˜ ˜ ˜ ˜ ˜ Q23 − Q12 Q11 Q13 + K23 Q22 − Q12 Q11 Q12  <0 (4) T ˜ −1 ˜ T T ˜ −1 ˜ ˜T ˜ ˜ ˜ ˜ Q Q Q Q Q Q − Q + K − Q 12 33 13 23 13 11 23 13 11. June 16, 2003. DRAFT.

(6) 4. ˜ 11 , Q ˜ 12 , Q ˜ 13 and Q ˜ 33 are constant. Thus, we can According to condition (iii), the elements Q ˜ −1 ˜ ˜ 23 = −Q ˜ 23 + Q ˜T ˜ ˜ T ˜ −1 ˜ choose K 12 Q11 Q13 as a constant, which yields Q22 − Q12 Q11 Q12 < 0 and ˜ 33 − Q ˜ −1 ˜ ˜T ˜ Q 13 Q11 Q13 < 0 as remaining conditions together with Q11 < 0. These are equivalent to     ˜ ˜ ˜ ˜ Q Q Q Q  11 12  < 0  11 13  < 0 and (5) T ˜ ˜T ˜ ˜ Q Q Q Q 22 33 12 13 ˜ 11 (or equivalently T1 ) is which in turn is equivalent to condition (i) and (ii), respectively. If Q ˜ 23 = −Q ˜ 23 . If K ˜ 23 is not present (either T2 , T3 or both missing) only one not present, choose K ˜ 11 < 0 remains. of (5) or Q Finally, including the fourth row and column of (2), if present, we can always find a constant ˜ 44 = −σI provided conditions (i) and (ii) hold, by choosing σ > 0 sufficiently large. Here we K use the assumption that the set Q is closed. Condition (i) and (ii) correspond to the equivalent conditions of the standard elimination lemma, see for instance [4, Theorem 1]. We will now use this new result on a gain scheduling synthesis problem in the following ˙ : θ˙ ∈ Θd (θ)}, where Θd (θ) is defines a closed manner. For each valid θ ∈ Θ, let Qθ = {Q(θ, θ) ˙ We assume that 0 ∈ Θd (θ). Applying Lemma 1 on set, which bounds θ. ˙ + U (θ)K(θ)V T (θ) + V (θ)K T (θ)U T (θ) < 0 Q(θ, θ). (6). ˙ allows us to find conditions for the existence of K(θ) that is independent of θ. III. LPV S YNTHESIS The problem addressed here is the following. Suppose we are given a linear time-varying (LPV) plant with state-space realization x˙ = A(θ)x + B1 (θ)w + B2 (θ)u z = C1 (θ)x + D11 (θ)w + D12 (θ)u. (7). y = C2 (θ)x + D21 (θ)w + D22 (θ)u where A ∈ Rn×n , D11 ∈ Rp1 ×m1 and D22 ∈ Rp2 ×m2 define the problem dimension. The timevarying parameters θ ∈ Θ as well as its rate of variation, θ˙ ∈ Θd are bounded. We assume that Θ and Θd are closed sets. June 16, 2003. DRAFT.

(7) 5. The gain-scheduled output-feedback control problem consists of finding a dynamic LPV controller with state-space equations x˙ K = KA (θ)x + KB (θ)y. (8). u = KC (θ)x + KD (θ)y where KA ∈ Rr×r that ensures internal stability and a guaranteed performance bound, γ. The performance bound is defined as the L2 -induced norm of the closed loop system from disturbance input signal, w, to the performance output, z, that is Z T Z T T 2 z zdt ≤ γ wT wdt, 0. ∀T ≥ 0,. (9). 0. ˙ assuming zero initial states, x(0) = 0. which shall hold for all admissible trajectories (θ, θ) We use the index K to denote the state-space matrices of the closed loop system. In the ˙ There is no loss of generality by assuming sequel we drop the explicit dependency on θ and θ. D22 = 0. If K is the controller for D22 = 0, then the controller for D22 6= 0 is K(I + D22 K)−1 . Hence, we can write the closed loop system as         A 0 B1 B2 0      A B    KD KC   C2 0 D21   K K =  0 0 0  +  0 I      KB KA CK DK 0 I 0 C1 0 D11 D12 0. (10). where AK ∈ R(n+r)×(n+r) . The closed loop system is internally stable and has an L2 -induced norm of γ if there exists a symmetric, parameter-varying P = P T > 0 such that   T T ˙  P AK + AK P + P P BK CK   T T    BK P −γI DK   CK DK −γI   T        ˙ 0  P 0 P 0       AK BK   I 0 0    =  0 −γI 0  +  0 0   +  <0       CK DK 0 I 0 0 0 −γI 0 I holds, see [6, LTV Strict Bounded Real Lemma]. Inserting this in (11) together with    −1 X N Y M =  > 0. P = T T N L M ∗ June 16, 2003. (11). (12). DRAFT.

(8) 6. where N, M ∈ Rn×r yields   XA + AT X + X˙ AT N + N˙ XB1 C1T    N T A + N˙ T L˙ N T B1 0      T T T   X B N −γI D B 1 1 11   C1 0 D11 −γI    XB2 N         N TB L  K K 2    D C   C2 0 D21 0   + +   0  0  0 I 0 0   KB KA D12 0. T    <0 . (13). Using the standard elimination lemma [4, Theorem 1] or condition (i) and (ii)   (see for instance KD K C  is equivalent to of Lemma 1), the existence of K =  KB KA    T XA + AT X + X˙ XB C T   1 1   N 0 NX 0 T   X   <0 (14)   B1T X −γI D11   0 I 0 I C1 D11 −γI and . T. .   NY 0     0 I . AY + Y AT − Y˙ Y C1T B1 Y C1. −γI D11. B1T. T D11 −γI. .     NY 0  <0  0 I . where NX and NY designate any bases of the null spaces of. (15). i. h C2 D21. h and. i T , B2T D12. respectively. See [1] for details. A third LMI is required to link X and Y according to (12), see [5, Lemma 6.2] for details.. .  . X I I Y.  ≥ 0.. (16). Note that (16) is equivalent to X − Y −1 ≥ 0. Next, applying condition (iii) of Lemma 1 on (13),. June 16, 2003. DRAFT.

(9) 7. we obtain an additional sufficient condition for the existence of a K independent of X˙ and N˙ : ⊥T    XB N 2  T XA + AT X + X˙ AT N + N˙ XB C T   1 1    T NX 0 N B L 2   T T T      B1 X B1 N −γI D11      0 I 0    0 C1 0 D11 −γI D12 0   T XAY + AT + XY   ˙ + N˙ M T C1T XB1    NX 0  NY 0  T   = const. (17) =  B1T D11 −γI     0 I 0 I C1 Y −γI D11 where we have used  XB2 N   N TB L 2    0 0  D12 0. ⊥. .    0     =  0    D12. ⊥.       T  Y M 0 0   −1 I 0 NY 0   P =  0 0 0 I     0 I 0 I 0  0 0 I 0 0. B2 0. and XY + N M T = I,. (18). which is a consequent of (12). T ˙ + N˙ M T )NY 1 is constant for each θ ∈ Θ there exists According to condition (17), if NX1 (XY ˙ or equivalently, independent of X˙ and N˙ . Here NX1 and NY 1 a K(θ) that is independent of θ,. denote the upper n rows of of NX and NY respectively. The natural choice in most, if not all, applications is to to choose T ˙ + N˙ M T )NY 1 = 0 (XY NX1. (19). This condition is always satisfied if ˙ + N˙ M T = 0 XY. (20). holds. Note that condition (20) is not new, see for instance [1], [6], in which N and M were square, nonsingular matrices. The contribution of this paper is to show that conditions (18) and (20) apply even if N and M are nonsingular or not square. We will now show that for any N and M satisfying (18), it is possible to construct a P such that (12) holds provided X − Y −1 > 0 holds. June 16, 2003. DRAFT.

(10) 8. Lemma 2: Let X = X T , Y = Y T > 0 such that X − Y −1 > 0. Further assume that N and M are compatibly sized matrices such that XY + N M T = I. Then, there exists L = LT > 0 such that.  P =. Proof: Factor X − Y −1.  X N. . =. −1 Y. M. . >0 (21) N L M ∗ = N0 N0T , which is equivalent to N M T = I − XY = −N0 N0T Y = T. T. N0 M0T where M0 = −Y N0 . Let    −1 0 M0  X N0 0   Y  T   T  P0 =  N0 I 0  =  M0 I + N0T Y N0 0  > 0     0 0 I 0 0 I. (22). on which we perform a congruence transformation using   0  I   −1 T =  0 N0 N    T⊥ 0 M as post-multiplier. Note that T is nonsingular, since     −1 h i I ∗ N N  M T M −T M T⊥T =    0 0 T⊥ T⊥ T⊥T 0 M M M is nonsingular. Then indeed P = T T P0 T satisfies (21), since −1  −1      −1 −1 −1 T M N N N N N N   0 = 0   0  M = M.  0 T⊥ T⊥ T⊥ 0 M M M h This result together with the observation that by choosing N =. i X I ,M =. can always satisfy (18) and (20) provided X − Y −1 > 0 or, equivalently   X I  >0 I Y. (23) h. −Y I. i we. (24). holds. This is marginally more convervative than the non-strict (16). With this choice of N and M we can use.  L=.  X I I Y. June 16, 2003. . (X − Y −1 )−1 0 0. I.  .  X I. . (25). I Y DRAFT.

(11) 9. Note that condition (24) is also used in [1] and elsewhere. With this choice of N and M , the number of states in the resulting controller are twice as many as in the original system. We are now ready to summarize the main result of this paper. Theorem 1: Consider the LPV plant (7) governed by (8), with parameter trajectories con˙ strained by θ ∈ Θ and θ˙ ∈ Θd . Then there exist a θ-independent gain-scheduled output-feedback controller that gives internal stability and an L2 -induced norm bound by γ if there exist parameter˙ in Θ×Θd the infinite dimensional dependent matrices X(θ) and Y (θ) such that for all pairs (θ, θ) LMI defined by (14), (15) and (24) hold. IV. C ONCLUSIONS Sufficient conditions for the existence of a gain scheduled controller for LPV systems are presented. It is shown that it is always possible to find a practically valid controller, which depends on the parameters only and not on the rate of parameter variation. The conditions are three LMIs, which are similar to the conditions used for standard H∞ synthesis problems. In the general case, the controller needs to have twice as many states as the original system. If a lower-order controllers is searched for, a rate-independent controller still resists a convex formulation. R EFERENCES [1] P. Apkarian and R. Adams. Advancded gain-scheduling techniques for uncertain systems. IEEE Transactions on Control Systems Technology, 6(1):21–32, January 1998. [2] M. Chilali and P. Gahinet. H∞ design with pole placement constraints: An LMI approach. IEEE Transactions on Automatic Control, 41(3):358–367, March 1996. [3] G. E. Dullerud and F. Paganini. A Course in Robust Control Theory, volume 36 of Texts in Applied Mathematics. Springer, New York, 2000. [4] T. Iwasaki and R. E. Skelton. All controllers for the general H∞ control problem: LMI existence conditions and state space formulas. Automatica, 30:1307–1317, August 1994. [5] A. Packard. Gain scheduling via linear fractional transformations. Systems & Control Letters, 22(2):79–92, February 1994. [6] C. Scherer. Mixed H2 /H∞ control. In A. Isidori, editor, Trends in Control – A European Perspective, pages 173–216. Springer-Verlag, Rome, Italy, 1995.. June 16, 2003. DRAFT.

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