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The Non Interacting Control Problem for Linear Parameter Varying Systems
4th IFAC Symposium on System, Structure and Control Università Politecnica delle Marche Ancona, Italy, Sept 15-17, 2010
The Non Interacting Control Problem for Linear Parameter Varying Systems A. M. Perdon - A. Bonci DIIGA, Universit` a Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy e-mail: [email protected] - [email protected] Abstract: The Non Interacting Control Problem for linear parameter varying (LPV) system is investigated. Using a geometric approach, the notion of robust controllability subspace is introduced and computational aspects are discussed. Solvability conditions for the Non Interacting Control Problem by state feedback for LPV systems are obtained and procedures are given to practically compute the solutions, if any. Keywords: Linear Parameter Varying Systems, Non Interacting Control Problem, geometric approach. 1. INTRODUCTION
2. PRELIMINARIES AND NOTATIONS
The Non Interacting Control Problem consists, from an input–output point of view, in splitting a given system into k independent, output controllable subsystems by means of a state feedback. This paper is devoted to studying from a geometric point of view the Non Interacting Control Problem for a linear system depending on time-varying parameters. The Non Interacting Control Problem from a state space point of view has been solved for standard linear systems, using a geometric approach, in Basile Marro (1970) and Wohnam (1985). The geometric approach has been extended to parameter varying systems (see Basile Marro (1987), Conte et al. (1991), Bokor et al. (2002) and references therein) and it is based on suitable notions of robust invariant subspaces, which, together with the invariance with respect to the dynamics, capture the concept of robustness with respect to parameter variations. In this paper, after recalling the fundamental notion of robust controlled invariance, we define the geometric notion of maximum robust controllability subspace for linear system depending on parameters and discuss the computational aspect of the procedures to practically compute it in different situations. Then, we state the Robust Non Interacting Control Problem for LPV systems and we give solvability conditions in geometric terms, namely using the notion of maximum robust controllability subspace. The independence of the solution from the parameter variations motivates the choice of calling robust the solution we develop, although the compensated system itself depends on the parameter. A procedure to compute a solution, if any, is illustrated by an example. 978-3-902661-83-8/10/$20.00 © 2010 IFAC
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Let us consider a linear system Σ(ρ), depending on a vector of real parameters ρ = [ρ1 , ρ2 , ..., ρs ]T , ρ ∈ P ⊆ Rs , where P is an open set, defined by the the following equations x(t) ˙ = A(ρ)x(t) + B(ρ)u(t) Σ(ρ) : , (1) y(t) = C(ρ)x(t) x(·) ∈ X = Rn is the state vector, u(·) ∈ U = Rm is the input vector, y(·) ∈ Y = Rp is the output vector and A(ρ), B(ρ), C(ρ) are matrices of suitable dimensions with entries depending on ρ. Our goal is to investigate geometric conditions which guarantee the existence of a state feedback that decouples the system into k independent, output controllable subsystems. The key tool in a state space solution of the Non Interacting Control Problem are the notions of controlled invariant and of controllability subspaces (Basile Marro (1970)). Let us start by recalling the notion of robustness for controlled invariant subspaces. Definition 1. Given the system Σ(ρ) of the form (1), a subspace V ⊂ X = Rn is a robust controlled invariant subspace (Basile Marro (1987)) for Σ(ρ) with respect to P or robust (A, B)-invariant if the following relation hold: A(ρ)V ⊂ V + Im B(ρ) for all ρ ∈ P (2) A robust controlled invariant subspace exhibits the same properties of standard controlled invariant subspaces described in Basile Marro (1969), Basile Marro (1992) and in Wohnam (1985). In particular the following: Proposition 2. Given a subspace V ⊂ X = Rn and two real numbers t0 and t1 with t0 < t1 , for any initial state x(t0 ) and any ρ ∈ P there exists at least one control function u|[t0 ,t1 ] such that the corresponding state trajectory x|[t0 ,t1 ] completely belongs to V if and only if V is a robust controlled invariant subspace. Proposition 3. Given a subspace V ⊂ Rn , for any ρ ∈ P there exists a state feedback matrix F (ρ) such that 10.3182/20100915-3-IT-2017.00076
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[A(ρ) + B(ρ)F (ρ)]V ⊂ V
(3)
if and only if V is a robust controlled invariant subspace. Proposition 4. The set of robust controlled invariant subspaces of Σ(ρ) contained in a given subspace E ⊆ Rn is an upper semi–lattice with respect to the inclusion and the sum of subspaces. Then, it exists a maximum element, denoted by ∗ Vmax (A(ρ), B(ρ), E) ∗ or simply by V (E) or V ∗ , when no confusion may arise. An important aspect of the geometric approach is the practical computability of its basic objects. For the computation of the maximum robust controlled invariant subspace in a given space two algorithm have been proposed, concerning two different kind of parameter dependence of the system Σ(ρ). We recall them, since they are needed in the following. Affine case Assume that the parameter varying system Σ(ρ) is defined by equations of the form (1), where the defining matrices are affine in the parameters: A(ρ) = A0 + ρ1 A1 + . . . + ρs As B(ρ) = B0 + ρ1 B1 + . . . + ρs Bs (4) C(ρ) = C0 + ρ1 C1 + . . . + ρs Cs Assumption 1 Let us assume Im B(ρ) = B, for any ρ ∈ P
(5)
Remark that in Bokor et al. (2002) and Bokor Balas (2005) the more restrictive hypothesis B(ρ) = B for any ρ ∈ P is assumed. RISA-1 Robust controlled Invariant Subspace Algorithm. Let E be a fixed subspace of X = Rn and consider the following sequence of subspaces : V0 = E \ \ (6) Vk+1 = E A−1 i (Vk + B) i=0,..,s
The sequence {V0 , . . . , Vk , . . .} converges in a finite number of steps to V ∗ (E). Polynomial case Assumption 2 Assume that Σ(ρ) is defined by equations of the form (1), where P is an open subset of R and the elements of A(ρ), B(ρ) and C(ρ) are polynomials in the scalar parameter ρ ∈ P. Let us introduce the following notation: given a family W (ρ) of subspaces of Rn depending on the parameter ρ ∈ P, we denote by W (ρ) the intersection of all the elements of the family, i.e. \ W (ρ) = W (ρ) (7) ρ∈P
If Assumption 2 is satisfied, W (ρ) can be practically computed by means of the algorithm described in Conte et al. (1991) Section 2.2. 193
RISA-2 Robust controlled Invariant Subspace Algorithm. Let E be a fixed subspace of X = Rn and consider the following sequence of subspaces : V0 = E \ (8) Vk+1 = Vk A(ρ)−1 (Vk + Im B(ρ)) The sequence {V0 , . . . , Vk , . . .} converges in a finite number of steps to V ∗ (E). 3. THE ROBUST CONTROLLABILITY SUBSPACE In order to deal with the Non Interacting Control Problem for LPV systems we have to introduce another geometric notion, that of robust controllability subspace and investigate its properties. Definition 5. Given a system Σ(ρ) of the form (1), a subspace R ⊆ X = Rn is a robust controllability subspace if for each ρ ∈ P there exist linear maps F (ρ) : X → U and G(ρ) : U → U such that
R = < [A(ρ) + B(ρ)F (ρ)] | Im [B(ρ)G(ρ)] >
(9)
As it happens for standard systems, we can show the following: Proposition 6. A subspace R ⊂ X is a robust controllability subspace for Σ(ρ) if and only if for each ρ ∈ P there exists a linear map F (ρ) : X → U such that \ R = < [A(ρ) + B(ρ)F (ρ)] | Im B(ρ) R > (10) Proposition 7. The set of robust controllability subspaces contained in a given E ⊆ Rn is an upper semi–lattice with respect to the inclusion and the sum of subspaces. Then, there exists a maximum element, denoted by R∗max (A(ρ), B(ρ), E)
(11)
or simply by R∗ (E), when no confusion may arise. A robust controllability subspace is obviously a robust controlled invariant subspace since it satisfies (3). Moreover, the family of robust controllability subspaces of a fixed pair (A(ρ), B(ρ)) is a subfamily of that of the robust controlled invariant subspaces whose importance derives from the fact that the restriction of (A(ρ) + B(ρ)F (ρ)) to a controllability subspace can be assigned arbitrary by a suitable choice of F (ρ). Let E be a fixed subspace of X = Rn and denote by V ∗ the maximum robust controlled invariant subspace in E. In this section we present two algorithm for computing in a finite number of steps the maximum robust controllability subspace contained in a given E ⊆ X . Here again we have different algorithms for linear systems depending differently on the parameters. Affine case Assume that Σ(ρ) is defined by equations of the form (1), where the defining matrices are affine in the parameters, i.e. satisfy (4) and that Assumption 1 with (5) holds.
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RCSA-1 Robust Controllability Subspace Algorithm. Step 1 Compute V ∗ (E) by the algorithm RISA-1, equations (6) Step 2 Consider the following sequence of subspaces R0 = 0 ! s \ X (12) ∗ Rk+1 = V (E) Ai Rk + B i=1
Proposition 8. The sequence of subspaces {R0 , . . . , Rk , . . .} defined by (12) converges in a finite number of steps to R∗ (E), the maximum robust controllability subspace contained in E. Sketch of the proof At any step we have the intersection of a finite number of subspaces. The algorithm generates a non decreasing sequence of subspaces R0 ⊆ R1 ⊆ . . . whose dimension at any step is lesser by at least one or stabilizes. Then there exists an integer r such that Rr+1 = Rr . Polynomial case Assume that Σ(ρ) is defined by equations of the form (1), where P is an open subset of R and the elements of A(ρ), B(ρ) and C(ρ) are polynomials in the scalar parameter ρ ∈ P.
behavior of the remaining yj , j 6= i. This is called also a Block Decoupling Control and such a system is called decoupled. In general it is not possible to decouple a system as described above, but the goal can be achieved by introducing a state feedback, and by regrouping the input control variables in suitable combinations. A further objective is to cancel or to compensate inherent cross-couplings and to achieve a satisfactory dynamic response. Problem formulation: Let us rewrite the LPV system Σ(ρ) described by equations (1) according to (14) as x(t) ˙ = A(ρ)x(t) + B(ρ)u(t) ˜ Σ(ρ) , (15) yi (t) = Ci (ρ)x(t) i = 1, . . . , k where each block output yi belongs to the i-th output subspace Yi = Rpi with i = 1, . . . , k C1 (ρ) C (ρ) C(ρ) = 2 ... Ck (ρ) and A(ρ), B(ρ), Ci (ρ), are matrices of suitable dimensions with entries depending on ρ. Therefore Y = Y1 ⊕ . . . ⊕ Yk . ˜ Associated to Σ(ρ) let us consider the static state feedback law
RCSA-2 Robust Controllability Subspace Algorithm. Step 1 Compute V ∗ (E) by algorithm algorithm RISA-1, equations (8) Step 2 Consider the following sequence of subspaces R0 = 0 \ (13) Rk+1 = V ∗ (E) A(ρ)Rk + Im B(ρ) Since Assumption 2 is satisfied, A(ρ)Rk + Im B(ρ) can be practically computed by means of the algorithm described in Conte et al. (1991) Section 2.2. The non decreasing sequence {R0 , . . . , Rk , . . .} converges in a finite number of steps to R∗ (E). 4. THE NON INTERACTING CONTROL PROBLEM FOR LPV SYSTEMS The Non Interacting Control Problem (NICP) is an important problem in several applications, in particular in dealing with complex systems. Consider a multi-variable LPV system Σ(ρ) defined by equations (1), whose output y(·) has been divided in k, k ≤ m disjoint subsets, each having a physical significance to distinguish it from the remaining subsets. Represent the output subsets by vectors X yi = [yi1 , ..., yipi ]T i = 1, 2, . . . , k, pi = p (14) i=1,k
Next suppose to partition the m-dimensional input u(·) of the system into k disjoint subsets X ui = [ui1 , ..., uimi ]T i = 1, 2, . . . , k, mi = m i=1,k
such that for each i = 1, 2, . . . , k the inputs of ui completely control the output vector yi without affecting the 194
u(t) = F (ρ)x(t) +
k X
Gi (ρ)vi (t),
(16)
i=1
where the vi (t) are new inputs and F (ρ) and Gi (ρ), i = 1, 2, . . . , k, are matrices of suitable dimensions with entries depending on ρ. ˜ The Non Interacting Control Problem for Σ(ρ) consists in finding, if possible, a static state feedback law (16) such that in the compensated system x(t) ˙ = [A(ρ) + B(ρ)F (ρ)]x(t)+ k X ˜ c (ρ) Σ (17) B(ρ)Gi (ρ)vi (t) i=1 yi (t) = Ci (ρ)x(t) i = 1, 2, . . . , k each block input vi (·) completely controls the corresponding block output yi (·) but has no influence on yj (·) for i 6= j. ˜ c (ρ) that are The states of the compensated system Σ reachable by the input vi are those belonging to the subspace
Ri (ρ) = < [A(ρ) + B(ρ)F (ρ)] | Im [B(ρ)Gi (ρ)] > (18) namely, to the the controllability subspace generated by the control input vi .
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The non–interaction condition requiring that in the compensated system the i-th input vi does not affect the outputs yj (·), j 6= i can be formulated as follows: Cj (ρ)Ri (ρ) = 0,
j 6= i,
i, j = 1, 2, . . . , k, ∀ρ ∈ P (19)
The output controllability condition requiring that, at the same time, vi completely controls yi , is verified if any element in the image of Ci (ρ) can be reached by a suitable choice of vi . The algebraic formulation of this condition is then Ci (ρ)Ri (ρ) = Im Ci (ρ), i = 1, 2, . . . , k,
∀ρ ∈ P
(20)
Then, the Non Interacting Control Problem can be formulated as follows. ˜ Given the system Σ(ρ), find (if possible) F (ρ) and Gi (ρ), i = 1, 2, . . . , k, such that the subspaces Ri (ρ) defined by (18) verify conditions (19) and (20). Let us denote by Ki (ρ) =
k \
Ker Cj (ρ)
i = 1, 2, . . . , k, ∀ρ ∈ P (21)
j6=i
Then, conditions (19) and (20) respectively can be restated as Ri (ρ) ⊆ Ki (ρ)
i = 1, 2, . . . , k, ∀ρ ∈ P (22)
Ri (ρ) + Ker Ci (ρ) = X
i = 1, 2, . . . , k, ∀ρ ∈ P. (23)
Summarizing, the Robust Non Interacting Control Problem ˜ (RNICP) for Σ(ρ) can be stated as follows: find, if possible, a static feedback law (16) and robust controllability subspaces Ri , i = 1, 2, . . . , k, such that equations (22) and (23) are verified. Then, the Robust Non Interacting Control Problem amounts to seeking robust controllability subspace which are small enough to guarantee non–interaction, yet large enough to ensure output controllability. 5. SOLUTION OF THE NON INTERACTING CONTROL PROBLEM FOR LPV SYSTEMS ˜ Proposition 9. Consider the system Σ(ρ) described by equations (15) and assume that the pair (A(ρ), B(ρ)) is controllable for each value of the vector of real parameters ρ = [ρ1 , ρ2 , ..., ρs ]T , ρ ∈ P ⊆ Rs , furthermore assume that Ki (ρ), defined as in (21), satisfy the relations Ker Ki (ρ) = Ki
for all ρ ∈ P
(24)
Then, denoting by R∗i the maximum robust controllability subspace contained in Ki , the Robust Non Interacting ˜ Control Problem (RNICP) for Σ(ρ) is solvable if and only if for all ρ ∈ P the following relations hold R∗i + Ker Ci (ρ) = X k \
i = 1, . . . , k.
Ker Ci (ρ) = 0
(25) (26)
Condition (25) derives directly from non interacting and controllability conditions. In fact, given the {Ri } of the form (18) and the Ki , if R∗i is the maximum robust controllability subspace contained in Ki , necessary it satisfy equation (19). If there exist R∗i which satisfy (25) for each i = 1, . . . , k then it satisfy equation (23). Conversely, if for some i, R∗i fails to satisfy (23), then clearly the Robust Non Interacting Control Problem (RNICP) is not solvable, since R∗i is the maximum. If for each i = 1, . . . , k, R∗i satisfies (23) it might exist a family of controllability subspaces R1 . . . Rk for which Ri ⊂ R∗i with all Ri large enough to satisfy (23). Proof of Proposition 9 Necessity. Assume that the Robust Non Interacting Control Problem (RNICP) is solvable and, for every i = 1, . . . , k, let Ni (ρ) =< A(ρ) + B(ρ)F (ρ) | Im [B(ρ)Gi (ρ)] > be the subspace of the states of the closed loop system Σ˜c that are reachable using the i-th input vi . By the output controllability condition (23), for every i we have, Ni + KerCi = X and, by the non interaction condition (22), Ni ⊆ Ki . Then, Ni ⊆ R∗i and R∗i + Ker Ci = X . Sufficency. By (25), any x ∈ X can be written as x = r1 + c1 , with r1 ∈ R∗1 and c1 ∈ Ker C1 . Using (25) again, c1 can ∗ be written as c1 = r2 + c2 , with r2 ∈ RT 2 and c2 ∈ Ker C2 . Then x = r1 +r2 +c2 with c2 ∈ Ker C1 Ker C2 . Iterating this procedure, at step k we get x = r1 + r2 + . . . + rk + ck , Tk with ri ∈ R∗i and ck ∈ i=1 Ker Ci . Therefore ck = 0 by ∗ ∗ (26) and R1 + R2 + . . . + R∗k = X . Let us show that the above sum is direct. Assume that x can also be written as x = r1′ + r2′ + . . . + rk′ , with ri′ ∈ R∗i . Subtracting the two expressions, we have 0 = r1 − r1′ + . . . + rk − rk′ , then Pk ri − ri′ = j=1,j6=i (rj′ − rj ) belongs both to R∗i ⊆ Ki and to KerCi . Therefore ri − ri′ = 0 for every i = 1, . . . , k and X = R∗1 ⊕ R∗2 ⊕ . . . ⊕ R∗k .
For every i = 1, . . . , k R∗i is a robust controllability submodule, then there exist maps Fi (ρ), Gi (ρ) such that R∗i =< A(ρ) + B(ρ)Fi (ρ) | Im B(ρ) ∩ R∗i > for every ρ ∈ P. Expressing every element of X as x = x1 + x2 + . . . + xk , xi ∈ R∗i , the feedback F (ρ) defined by F (ρ)x = F1 (ρ)x1 + F2 (ρ)x2 + . . . + Fk (ρ)x T k is such that R∗i =< A(ρ) + B(ρ)F (ρ) | Im B(ρ) R∗i > for i = 1, . . . , k, for every ρ ∈ P. Letting {ui1 . . . , uimi } be a basis mi of the subspace B −1 (ρ)(R∗i ) ⊆ U, define T Gi (ρ) : R → U by Gi (ρ)(ej ) = uij . Then, Im B(ρ) Ri = ImB(ρ)Gi (ρ) and {F (ρ), G1 (ρ), . . . , Gk (ρ)} is a solution of the RNICP for Σ.
6. EXAMPLE Let us consider a parameter vector ρ = [ρ1 , ρ2 ]T in the open set P ⊆ R2 , where P = {ρ1 , ρ2 : ρ1 ρ2 − 1 6= 0}. Let us consider the system Σ given by x˙ (t) = x1 (t) + ρ2 x2 (t) + u1 (t) + ρ1 u2 (t) 1 x˙ 2 (t) = (ρ1 + ρ2 )x1 (t) + ρ2 x2 (t) + u2 (t) Σ= y1 (t) = ρ1 x1 (t) y2 (t) = ρ2 x2 (t) The matrices describing the system Σ are:
i=1
195
(27)
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A(ρ) =
1 ρ2 1 ρ1 , B(ρ) = , (ρ1 + ρ2 ) ρ1 ρ2 1 ρ1 0 C(ρ) = 0 ρ2
the system is controllable for each ρ ∈ P. Let us divide the output vector and the output matrix C1 (ρ) C(ρ) as follows, C(ρ) = , so that k=2 and C2 (ρ) C1 (ρ) = [ρ1 0] and C2 (ρ) = [0 ρ2 ]. We have that Im B(ρ) = B = R2 for every ρ ∈ P, then Assumption 1 is satisfied. The subspaces Ki (ρ), defined in (21), for i = 1, 2 are 1 K1 (ρ) = Ker C2 (ρ) = span{ } 0 0 K2 (ρ) = Ker C1 (ρ) = span{ } 1
Finally, the static feedback law u(t) = F (ρ)x(t) + G1 (ρ)v1 (t) + G2 (ρ)v2 (t) where G1 and G2 are given by (28), achieves the wanted decoupling. In fact, 1 0 A(ρ) − B(ρ)F (ρ) = 0 ρ1 and the compensated system is x˙ (t) = x1 (t) + (ρ1 ρ2 − 1)v1 1 x˙ 2 (t) = ρ1 x2 (t) + (ρ1 ρ2 − 1)v2 (t) Σ= y1 (t) = ρ1 x1 (t) y2 (t) = ρ2 x2 (t)
so that (26) holds. Since they are independent on ρ, also condition (24) is satisfied. Then, by using algorithms CRSA-1 we can compute the maximum robust controllability subspaces contained in Ki , i = 1, 2. We obtain that R∗1 and R∗2 are the subspaces spanned by the matrices " # " # ρ1 ρ2 − 1 0 R1 (ρ) = , R2 (ρ) = 0 ρ1 ρ2 − 1 respectively.
that is in fact decoupled. Remark that the problem is solvable only if the open set P to which the parameter ρ belongs does not contain the curve ρ1 ρ2 −1 = 0. In fact, for the parameter value (ρ1 , ρ2 ) that lie on this curve R∗1 = R∗2 = 0 and condition (25) is not satisfied. 7. CONCLUSIONS
From (9) and (10) we can compute the map Gi (ρ), i = 1, 2 such that \ R∗i B = Im [B(ρ)Gi (ρ)] by solving the equation B(ρ)Gi (ρ) = Ri and than we have " # " # −1 ρ1 G1 (ρ) = , G2 (ρ) = ρ2 −1
ρ (ρ + ρ ) ρ2 1 1 2 − ρ1 ρ2 − 1 ρ1 ρ2 − 1 F (ρ) = ρ1 + ρ2 ρ2 2 − ρ1 ρ2 − 1 ρ1 ρ2 − 1 satisfies the relations [(A(ρ) + B(ρ)F (ρ)]R∗i ⊆ R∗i , i = 1, 2.
(28)
Since R∗1 and R∗2 are robust controlled invariant subspaces, they satisfy (2) hence we can solve the equations A(ρ)Ri (ρ) = Ri (ρ)Li (ρ) + B(ρ)Mi (ρ), i = 1, 2 and find Li (ρ) and Mi (ρ). Then, we can find matrices Fi (ρ) such that Fi (ρ)Ri = Mi . We obtain ρ1 (ρ1 + ρ2 ) ρ1 ρ2 − 1 f12 F1 (ρ) = ρ1 + ρ2 − f22 ρ1 ρ2 − 1 ρ2 f11 − ρ1 ρ2 − 1 F2 (ρ) = ρ2 2 f21 ρ1 ρ2 − 1 where the terms fij , i = 1, 2 and j = 1, 2 are freely assignable. Then, 196
In this work we have defined the geometric notion of robust controllability subspace for linear system depending on parameters and we have presented two different algorithms to compute the maximum controllability subspace contained in a given space for linear systems with an affine dependence on the parameters and for linear systems with a polynomial dependence on a parameter. We have stated the Robust Non Interacting Control Problem for LPV systems and we have given solvability conditions using the notion of maximum robust controllability subspace. A procedure to practically check the solvability conditions and to construct the solution, if any exist, is illustrated by an example. REFERENCES G. Basile, G. Marro, Controlled and conditioned invariant subspaces in linear system theory, J. Optim. Theory Appl. 5 (1969) pp 305 – 315 G. Basile, G. Marro, A state space approach to noninteracting controls, Ricerche di Automatica, 1 , n. 1, 1970. M. Wohnam, Linear multivariable control : a geometric approach, 3rd Ed. Springer Verlag, Berlin 1985; G. Basile, G. Marro, On the robust controlled invariant, System and Control Letters, 9, 1987. G. Conte, A. M. Perdon, G. Marro, Computing the maximum robust controlled invariant subspace, System and Control Letters, 17, pp. 131-136, 1991. J. Bokor, Z. Szabo and G. Stikkel, Invariant subspaces for LPV systems and their applications, Proc. Mediterranean Control Conference 2002, Lisbon, Portugal, 2002.
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J. Bokor, G. Balas, Linear Parameter Varying Systems: A Geometric Theory and Applications, Proc. 16th IFAC World Congress Prague,Czech Republic, 2005. G. Basile, G. Marro, Controlled and conditioned invariants in linear system theory, Prentice Hall, New York, 1992.
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The Non Interacting Control Problem for Linear Parameter Varying Systems A. M. Perdon - A. Bonci DIIGA, Universit` a Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy e-mail: [email protected] - [email protected] Abstract: The Non Interacting Control Problem for linear parameter varying (LPV) system is investigated. Using a geometric approach, the notion of robust controllability subspace is introduced and computational aspects are discussed. Solvability conditions for the Non Interacting Control Problem by state feedback for LPV systems are obtained and procedures are given to practically compute the solutions, if any. Keywords: Linear Parameter Varying Systems, Non Interacting Control Problem, geometric approach. 1. INTRODUCTION
2. PRELIMINARIES AND NOTATIONS
The Non Interacting Control Problem consists, from an input–output point of view, in splitting a given system into k independent, output controllable subsystems by means of a state feedback. This paper is devoted to studying from a geometric point of view the Non Interacting Control Problem for a linear system depending on time-varying parameters. The Non Interacting Control Problem from a state space point of view has been solved for standard linear systems, using a geometric approach, in Basile Marro (1970) and Wohnam (1985). The geometric approach has been extended to parameter varying systems (see Basile Marro (1987), Conte et al. (1991), Bokor et al. (2002) and references therein) and it is based on suitable notions of robust invariant subspaces, which, together with the invariance with respect to the dynamics, capture the concept of robustness with respect to parameter variations. In this paper, after recalling the fundamental notion of robust controlled invariance, we define the geometric notion of maximum robust controllability subspace for linear system depending on parameters and discuss the computational aspect of the procedures to practically compute it in different situations. Then, we state the Robust Non Interacting Control Problem for LPV systems and we give solvability conditions in geometric terms, namely using the notion of maximum robust controllability subspace. The independence of the solution from the parameter variations motivates the choice of calling robust the solution we develop, although the compensated system itself depends on the parameter. A procedure to compute a solution, if any, is illustrated by an example. 978-3-902661-83-8/10/$20.00 © 2010 IFAC
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Let us consider a linear system Σ(ρ), depending on a vector of real parameters ρ = [ρ1 , ρ2 , ..., ρs ]T , ρ ∈ P ⊆ Rs , where P is an open set, defined by the the following equations x(t) ˙ = A(ρ)x(t) + B(ρ)u(t) Σ(ρ) : , (1) y(t) = C(ρ)x(t) x(·) ∈ X = Rn is the state vector, u(·) ∈ U = Rm is the input vector, y(·) ∈ Y = Rp is the output vector and A(ρ), B(ρ), C(ρ) are matrices of suitable dimensions with entries depending on ρ. Our goal is to investigate geometric conditions which guarantee the existence of a state feedback that decouples the system into k independent, output controllable subsystems. The key tool in a state space solution of the Non Interacting Control Problem are the notions of controlled invariant and of controllability subspaces (Basile Marro (1970)). Let us start by recalling the notion of robustness for controlled invariant subspaces. Definition 1. Given the system Σ(ρ) of the form (1), a subspace V ⊂ X = Rn is a robust controlled invariant subspace (Basile Marro (1987)) for Σ(ρ) with respect to P or robust (A, B)-invariant if the following relation hold: A(ρ)V ⊂ V + Im B(ρ) for all ρ ∈ P (2) A robust controlled invariant subspace exhibits the same properties of standard controlled invariant subspaces described in Basile Marro (1969), Basile Marro (1992) and in Wohnam (1985). In particular the following: Proposition 2. Given a subspace V ⊂ X = Rn and two real numbers t0 and t1 with t0 < t1 , for any initial state x(t0 ) and any ρ ∈ P there exists at least one control function u|[t0 ,t1 ] such that the corresponding state trajectory x|[t0 ,t1 ] completely belongs to V if and only if V is a robust controlled invariant subspace. Proposition 3. Given a subspace V ⊂ Rn , for any ρ ∈ P there exists a state feedback matrix F (ρ) such that 10.3182/20100915-3-IT-2017.00076
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[A(ρ) + B(ρ)F (ρ)]V ⊂ V
(3)
if and only if V is a robust controlled invariant subspace. Proposition 4. The set of robust controlled invariant subspaces of Σ(ρ) contained in a given subspace E ⊆ Rn is an upper semi–lattice with respect to the inclusion and the sum of subspaces. Then, it exists a maximum element, denoted by ∗ Vmax (A(ρ), B(ρ), E) ∗ or simply by V (E) or V ∗ , when no confusion may arise. An important aspect of the geometric approach is the practical computability of its basic objects. For the computation of the maximum robust controlled invariant subspace in a given space two algorithm have been proposed, concerning two different kind of parameter dependence of the system Σ(ρ). We recall them, since they are needed in the following. Affine case Assume that the parameter varying system Σ(ρ) is defined by equations of the form (1), where the defining matrices are affine in the parameters: A(ρ) = A0 + ρ1 A1 + . . . + ρs As B(ρ) = B0 + ρ1 B1 + . . . + ρs Bs (4) C(ρ) = C0 + ρ1 C1 + . . . + ρs Cs Assumption 1 Let us assume Im B(ρ) = B, for any ρ ∈ P
(5)
Remark that in Bokor et al. (2002) and Bokor Balas (2005) the more restrictive hypothesis B(ρ) = B for any ρ ∈ P is assumed. RISA-1 Robust controlled Invariant Subspace Algorithm. Let E be a fixed subspace of X = Rn and consider the following sequence of subspaces : V0 = E \ \ (6) Vk+1 = E A−1 i (Vk + B) i=0,..,s
The sequence {V0 , . . . , Vk , . . .} converges in a finite number of steps to V ∗ (E). Polynomial case Assumption 2 Assume that Σ(ρ) is defined by equations of the form (1), where P is an open subset of R and the elements of A(ρ), B(ρ) and C(ρ) are polynomials in the scalar parameter ρ ∈ P. Let us introduce the following notation: given a family W (ρ) of subspaces of Rn depending on the parameter ρ ∈ P, we denote by W (ρ) the intersection of all the elements of the family, i.e. \ W (ρ) = W (ρ) (7) ρ∈P
If Assumption 2 is satisfied, W (ρ) can be practically computed by means of the algorithm described in Conte et al. (1991) Section 2.2. 193
RISA-2 Robust controlled Invariant Subspace Algorithm. Let E be a fixed subspace of X = Rn and consider the following sequence of subspaces : V0 = E \ (8) Vk+1 = Vk A(ρ)−1 (Vk + Im B(ρ)) The sequence {V0 , . . . , Vk , . . .} converges in a finite number of steps to V ∗ (E). 3. THE ROBUST CONTROLLABILITY SUBSPACE In order to deal with the Non Interacting Control Problem for LPV systems we have to introduce another geometric notion, that of robust controllability subspace and investigate its properties. Definition 5. Given a system Σ(ρ) of the form (1), a subspace R ⊆ X = Rn is a robust controllability subspace if for each ρ ∈ P there exist linear maps F (ρ) : X → U and G(ρ) : U → U such that
R = < [A(ρ) + B(ρ)F (ρ)] | Im [B(ρ)G(ρ)] >
(9)
As it happens for standard systems, we can show the following: Proposition 6. A subspace R ⊂ X is a robust controllability subspace for Σ(ρ) if and only if for each ρ ∈ P there exists a linear map F (ρ) : X → U such that \ R = < [A(ρ) + B(ρ)F (ρ)] | Im B(ρ) R > (10) Proposition 7. The set of robust controllability subspaces contained in a given E ⊆ Rn is an upper semi–lattice with respect to the inclusion and the sum of subspaces. Then, there exists a maximum element, denoted by R∗max (A(ρ), B(ρ), E)
(11)
or simply by R∗ (E), when no confusion may arise. A robust controllability subspace is obviously a robust controlled invariant subspace since it satisfies (3). Moreover, the family of robust controllability subspaces of a fixed pair (A(ρ), B(ρ)) is a subfamily of that of the robust controlled invariant subspaces whose importance derives from the fact that the restriction of (A(ρ) + B(ρ)F (ρ)) to a controllability subspace can be assigned arbitrary by a suitable choice of F (ρ). Let E be a fixed subspace of X = Rn and denote by V ∗ the maximum robust controlled invariant subspace in E. In this section we present two algorithm for computing in a finite number of steps the maximum robust controllability subspace contained in a given E ⊆ X . Here again we have different algorithms for linear systems depending differently on the parameters. Affine case Assume that Σ(ρ) is defined by equations of the form (1), where the defining matrices are affine in the parameters, i.e. satisfy (4) and that Assumption 1 with (5) holds.
SSSC 2010 Ancona, Italy, Sept 15-17, 2010
RCSA-1 Robust Controllability Subspace Algorithm. Step 1 Compute V ∗ (E) by the algorithm RISA-1, equations (6) Step 2 Consider the following sequence of subspaces R0 = 0 ! s \ X (12) ∗ Rk+1 = V (E) Ai Rk + B i=1
Proposition 8. The sequence of subspaces {R0 , . . . , Rk , . . .} defined by (12) converges in a finite number of steps to R∗ (E), the maximum robust controllability subspace contained in E. Sketch of the proof At any step we have the intersection of a finite number of subspaces. The algorithm generates a non decreasing sequence of subspaces R0 ⊆ R1 ⊆ . . . whose dimension at any step is lesser by at least one or stabilizes. Then there exists an integer r such that Rr+1 = Rr . Polynomial case Assume that Σ(ρ) is defined by equations of the form (1), where P is an open subset of R and the elements of A(ρ), B(ρ) and C(ρ) are polynomials in the scalar parameter ρ ∈ P.
behavior of the remaining yj , j 6= i. This is called also a Block Decoupling Control and such a system is called decoupled. In general it is not possible to decouple a system as described above, but the goal can be achieved by introducing a state feedback, and by regrouping the input control variables in suitable combinations. A further objective is to cancel or to compensate inherent cross-couplings and to achieve a satisfactory dynamic response. Problem formulation: Let us rewrite the LPV system Σ(ρ) described by equations (1) according to (14) as x(t) ˙ = A(ρ)x(t) + B(ρ)u(t) ˜ Σ(ρ) , (15) yi (t) = Ci (ρ)x(t) i = 1, . . . , k where each block output yi belongs to the i-th output subspace Yi = Rpi with i = 1, . . . , k C1 (ρ) C (ρ) C(ρ) = 2 ... Ck (ρ) and A(ρ), B(ρ), Ci (ρ), are matrices of suitable dimensions with entries depending on ρ. Therefore Y = Y1 ⊕ . . . ⊕ Yk . ˜ Associated to Σ(ρ) let us consider the static state feedback law
RCSA-2 Robust Controllability Subspace Algorithm. Step 1 Compute V ∗ (E) by algorithm algorithm RISA-1, equations (8) Step 2 Consider the following sequence of subspaces R0 = 0 \ (13) Rk+1 = V ∗ (E) A(ρ)Rk + Im B(ρ) Since Assumption 2 is satisfied, A(ρ)Rk + Im B(ρ) can be practically computed by means of the algorithm described in Conte et al. (1991) Section 2.2. The non decreasing sequence {R0 , . . . , Rk , . . .} converges in a finite number of steps to R∗ (E). 4. THE NON INTERACTING CONTROL PROBLEM FOR LPV SYSTEMS The Non Interacting Control Problem (NICP) is an important problem in several applications, in particular in dealing with complex systems. Consider a multi-variable LPV system Σ(ρ) defined by equations (1), whose output y(·) has been divided in k, k ≤ m disjoint subsets, each having a physical significance to distinguish it from the remaining subsets. Represent the output subsets by vectors X yi = [yi1 , ..., yipi ]T i = 1, 2, . . . , k, pi = p (14) i=1,k
Next suppose to partition the m-dimensional input u(·) of the system into k disjoint subsets X ui = [ui1 , ..., uimi ]T i = 1, 2, . . . , k, mi = m i=1,k
such that for each i = 1, 2, . . . , k the inputs of ui completely control the output vector yi without affecting the 194
u(t) = F (ρ)x(t) +
k X
Gi (ρ)vi (t),
(16)
i=1
where the vi (t) are new inputs and F (ρ) and Gi (ρ), i = 1, 2, . . . , k, are matrices of suitable dimensions with entries depending on ρ. ˜ The Non Interacting Control Problem for Σ(ρ) consists in finding, if possible, a static state feedback law (16) such that in the compensated system x(t) ˙ = [A(ρ) + B(ρ)F (ρ)]x(t)+ k X ˜ c (ρ) Σ (17) B(ρ)Gi (ρ)vi (t) i=1 yi (t) = Ci (ρ)x(t) i = 1, 2, . . . , k each block input vi (·) completely controls the corresponding block output yi (·) but has no influence on yj (·) for i 6= j. ˜ c (ρ) that are The states of the compensated system Σ reachable by the input vi are those belonging to the subspace
Ri (ρ) = < [A(ρ) + B(ρ)F (ρ)] | Im [B(ρ)Gi (ρ)] > (18) namely, to the the controllability subspace generated by the control input vi .
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The non–interaction condition requiring that in the compensated system the i-th input vi does not affect the outputs yj (·), j 6= i can be formulated as follows: Cj (ρ)Ri (ρ) = 0,
j 6= i,
i, j = 1, 2, . . . , k, ∀ρ ∈ P (19)
The output controllability condition requiring that, at the same time, vi completely controls yi , is verified if any element in the image of Ci (ρ) can be reached by a suitable choice of vi . The algebraic formulation of this condition is then Ci (ρ)Ri (ρ) = Im Ci (ρ), i = 1, 2, . . . , k,
∀ρ ∈ P
(20)
Then, the Non Interacting Control Problem can be formulated as follows. ˜ Given the system Σ(ρ), find (if possible) F (ρ) and Gi (ρ), i = 1, 2, . . . , k, such that the subspaces Ri (ρ) defined by (18) verify conditions (19) and (20). Let us denote by Ki (ρ) =
k \
Ker Cj (ρ)
i = 1, 2, . . . , k, ∀ρ ∈ P (21)
j6=i
Then, conditions (19) and (20) respectively can be restated as Ri (ρ) ⊆ Ki (ρ)
i = 1, 2, . . . , k, ∀ρ ∈ P (22)
Ri (ρ) + Ker Ci (ρ) = X
i = 1, 2, . . . , k, ∀ρ ∈ P. (23)
Summarizing, the Robust Non Interacting Control Problem ˜ (RNICP) for Σ(ρ) can be stated as follows: find, if possible, a static feedback law (16) and robust controllability subspaces Ri , i = 1, 2, . . . , k, such that equations (22) and (23) are verified. Then, the Robust Non Interacting Control Problem amounts to seeking robust controllability subspace which are small enough to guarantee non–interaction, yet large enough to ensure output controllability. 5. SOLUTION OF THE NON INTERACTING CONTROL PROBLEM FOR LPV SYSTEMS ˜ Proposition 9. Consider the system Σ(ρ) described by equations (15) and assume that the pair (A(ρ), B(ρ)) is controllable for each value of the vector of real parameters ρ = [ρ1 , ρ2 , ..., ρs ]T , ρ ∈ P ⊆ Rs , furthermore assume that Ki (ρ), defined as in (21), satisfy the relations Ker Ki (ρ) = Ki
for all ρ ∈ P
(24)
Then, denoting by R∗i the maximum robust controllability subspace contained in Ki , the Robust Non Interacting ˜ Control Problem (RNICP) for Σ(ρ) is solvable if and only if for all ρ ∈ P the following relations hold R∗i + Ker Ci (ρ) = X k \
i = 1, . . . , k.
Ker Ci (ρ) = 0
(25) (26)
Condition (25) derives directly from non interacting and controllability conditions. In fact, given the {Ri } of the form (18) and the Ki , if R∗i is the maximum robust controllability subspace contained in Ki , necessary it satisfy equation (19). If there exist R∗i which satisfy (25) for each i = 1, . . . , k then it satisfy equation (23). Conversely, if for some i, R∗i fails to satisfy (23), then clearly the Robust Non Interacting Control Problem (RNICP) is not solvable, since R∗i is the maximum. If for each i = 1, . . . , k, R∗i satisfies (23) it might exist a family of controllability subspaces R1 . . . Rk for which Ri ⊂ R∗i with all Ri large enough to satisfy (23). Proof of Proposition 9 Necessity. Assume that the Robust Non Interacting Control Problem (RNICP) is solvable and, for every i = 1, . . . , k, let Ni (ρ) =< A(ρ) + B(ρ)F (ρ) | Im [B(ρ)Gi (ρ)] > be the subspace of the states of the closed loop system Σ˜c that are reachable using the i-th input vi . By the output controllability condition (23), for every i we have, Ni + KerCi = X and, by the non interaction condition (22), Ni ⊆ Ki . Then, Ni ⊆ R∗i and R∗i + Ker Ci = X . Sufficency. By (25), any x ∈ X can be written as x = r1 + c1 , with r1 ∈ R∗1 and c1 ∈ Ker C1 . Using (25) again, c1 can ∗ be written as c1 = r2 + c2 , with r2 ∈ RT 2 and c2 ∈ Ker C2 . Then x = r1 +r2 +c2 with c2 ∈ Ker C1 Ker C2 . Iterating this procedure, at step k we get x = r1 + r2 + . . . + rk + ck , Tk with ri ∈ R∗i and ck ∈ i=1 Ker Ci . Therefore ck = 0 by ∗ ∗ (26) and R1 + R2 + . . . + R∗k = X . Let us show that the above sum is direct. Assume that x can also be written as x = r1′ + r2′ + . . . + rk′ , with ri′ ∈ R∗i . Subtracting the two expressions, we have 0 = r1 − r1′ + . . . + rk − rk′ , then Pk ri − ri′ = j=1,j6=i (rj′ − rj ) belongs both to R∗i ⊆ Ki and to KerCi . Therefore ri − ri′ = 0 for every i = 1, . . . , k and X = R∗1 ⊕ R∗2 ⊕ . . . ⊕ R∗k .
For every i = 1, . . . , k R∗i is a robust controllability submodule, then there exist maps Fi (ρ), Gi (ρ) such that R∗i =< A(ρ) + B(ρ)Fi (ρ) | Im B(ρ) ∩ R∗i > for every ρ ∈ P. Expressing every element of X as x = x1 + x2 + . . . + xk , xi ∈ R∗i , the feedback F (ρ) defined by F (ρ)x = F1 (ρ)x1 + F2 (ρ)x2 + . . . + Fk (ρ)x T k is such that R∗i =< A(ρ) + B(ρ)F (ρ) | Im B(ρ) R∗i > for i = 1, . . . , k, for every ρ ∈ P. Letting {ui1 . . . , uimi } be a basis mi of the subspace B −1 (ρ)(R∗i ) ⊆ U, define T Gi (ρ) : R → U by Gi (ρ)(ej ) = uij . Then, Im B(ρ) Ri = ImB(ρ)Gi (ρ) and {F (ρ), G1 (ρ), . . . , Gk (ρ)} is a solution of the RNICP for Σ.
6. EXAMPLE Let us consider a parameter vector ρ = [ρ1 , ρ2 ]T in the open set P ⊆ R2 , where P = {ρ1 , ρ2 : ρ1 ρ2 − 1 6= 0}. Let us consider the system Σ given by x˙ (t) = x1 (t) + ρ2 x2 (t) + u1 (t) + ρ1 u2 (t) 1 x˙ 2 (t) = (ρ1 + ρ2 )x1 (t) + ρ2 x2 (t) + u2 (t) Σ= y1 (t) = ρ1 x1 (t) y2 (t) = ρ2 x2 (t) The matrices describing the system Σ are:
i=1
195
(27)
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A(ρ) =
1 ρ2 1 ρ1 , B(ρ) = , (ρ1 + ρ2 ) ρ1 ρ2 1 ρ1 0 C(ρ) = 0 ρ2
the system is controllable for each ρ ∈ P. Let us divide the output vector and the output matrix C1 (ρ) C(ρ) as follows, C(ρ) = , so that k=2 and C2 (ρ) C1 (ρ) = [ρ1 0] and C2 (ρ) = [0 ρ2 ]. We have that Im B(ρ) = B = R2 for every ρ ∈ P, then Assumption 1 is satisfied. The subspaces Ki (ρ), defined in (21), for i = 1, 2 are 1 K1 (ρ) = Ker C2 (ρ) = span{ } 0 0 K2 (ρ) = Ker C1 (ρ) = span{ } 1
Finally, the static feedback law u(t) = F (ρ)x(t) + G1 (ρ)v1 (t) + G2 (ρ)v2 (t) where G1 and G2 are given by (28), achieves the wanted decoupling. In fact, 1 0 A(ρ) − B(ρ)F (ρ) = 0 ρ1 and the compensated system is x˙ (t) = x1 (t) + (ρ1 ρ2 − 1)v1 1 x˙ 2 (t) = ρ1 x2 (t) + (ρ1 ρ2 − 1)v2 (t) Σ= y1 (t) = ρ1 x1 (t) y2 (t) = ρ2 x2 (t)
so that (26) holds. Since they are independent on ρ, also condition (24) is satisfied. Then, by using algorithms CRSA-1 we can compute the maximum robust controllability subspaces contained in Ki , i = 1, 2. We obtain that R∗1 and R∗2 are the subspaces spanned by the matrices " # " # ρ1 ρ2 − 1 0 R1 (ρ) = , R2 (ρ) = 0 ρ1 ρ2 − 1 respectively.
that is in fact decoupled. Remark that the problem is solvable only if the open set P to which the parameter ρ belongs does not contain the curve ρ1 ρ2 −1 = 0. In fact, for the parameter value (ρ1 , ρ2 ) that lie on this curve R∗1 = R∗2 = 0 and condition (25) is not satisfied. 7. CONCLUSIONS
From (9) and (10) we can compute the map Gi (ρ), i = 1, 2 such that \ R∗i B = Im [B(ρ)Gi (ρ)] by solving the equation B(ρ)Gi (ρ) = Ri and than we have " # " # −1 ρ1 G1 (ρ) = , G2 (ρ) = ρ2 −1
ρ (ρ + ρ ) ρ2 1 1 2 − ρ1 ρ2 − 1 ρ1 ρ2 − 1 F (ρ) = ρ1 + ρ2 ρ2 2 − ρ1 ρ2 − 1 ρ1 ρ2 − 1 satisfies the relations [(A(ρ) + B(ρ)F (ρ)]R∗i ⊆ R∗i , i = 1, 2.
(28)
Since R∗1 and R∗2 are robust controlled invariant subspaces, they satisfy (2) hence we can solve the equations A(ρ)Ri (ρ) = Ri (ρ)Li (ρ) + B(ρ)Mi (ρ), i = 1, 2 and find Li (ρ) and Mi (ρ). Then, we can find matrices Fi (ρ) such that Fi (ρ)Ri = Mi . We obtain ρ1 (ρ1 + ρ2 ) ρ1 ρ2 − 1 f12 F1 (ρ) = ρ1 + ρ2 − f22 ρ1 ρ2 − 1 ρ2 f11 − ρ1 ρ2 − 1 F2 (ρ) = ρ2 2 f21 ρ1 ρ2 − 1 where the terms fij , i = 1, 2 and j = 1, 2 are freely assignable. Then, 196
In this work we have defined the geometric notion of robust controllability subspace for linear system depending on parameters and we have presented two different algorithms to compute the maximum controllability subspace contained in a given space for linear systems with an affine dependence on the parameters and for linear systems with a polynomial dependence on a parameter. We have stated the Robust Non Interacting Control Problem for LPV systems and we have given solvability conditions using the notion of maximum robust controllability subspace. A procedure to practically check the solvability conditions and to construct the solution, if any exist, is illustrated by an example. REFERENCES G. Basile, G. Marro, Controlled and conditioned invariant subspaces in linear system theory, J. Optim. Theory Appl. 5 (1969) pp 305 – 315 G. Basile, G. Marro, A state space approach to noninteracting controls, Ricerche di Automatica, 1 , n. 1, 1970. M. Wohnam, Linear multivariable control : a geometric approach, 3rd Ed. Springer Verlag, Berlin 1985; G. Basile, G. Marro, On the robust controlled invariant, System and Control Letters, 9, 1987. G. Conte, A. M. Perdon, G. Marro, Computing the maximum robust controlled invariant subspace, System and Control Letters, 17, pp. 131-136, 1991. J. Bokor, Z. Szabo and G. Stikkel, Invariant subspaces for LPV systems and their applications, Proc. Mediterranean Control Conference 2002, Lisbon, Portugal, 2002.
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J. Bokor, G. Balas, Linear Parameter Varying Systems: A Geometric Theory and Applications, Proc. 16th IFAC World Congress Prague,Czech Republic, 2005. G. Basile, G. Marro, Controlled and conditioned invariants in linear system theory, Prentice Hall, New York, 1992.
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