Disturbance Decoupling Problem

DISC Systems and Control Theory of Nonlinear Systems, 21 1 Disturbance Decoupling Problem Lecture 4 Nonlinear Dynamical Control Systems, Chapter 7 The disturbance decoupling problem is a typical example of a structural control problem which can be elegantly solved by geometric methods

DISC Systems and Control Theory of Nonlinear Systems, 21 2 Consider the nonlinear system affected by disturbances d ẋ = f(x) + g(x)u + e(x)d, u R m, d R l, y = h(x), y R p, on the state space manifold X Theorem 1 Suppose the system is analytic The disturbance d does not influence the output y, irrespectively of the initial condition x() = x and input u, if and only if there exists a distribution D on X such that

DISC Systems and Control Theory of Nonlinear Systems, 21 3 (i) (ii) (iii) D is invariant for ẋ = f(x) + g(x)u D ker dh e j D, j = 1,, l Recall that a distribution D is invariant for a vector field f if [f, X] D, X D or in shorthand notation [f, D] D D is invariant for ẋ = f(x) + g(x)u if [f, D] D, [g j, D] D, j = 1,, m

DISC Systems and Control Theory of Nonlinear Systems, 21 4 Necessity Take D = ker do where O is the observation space Sufficiency Assume that D is constant-dimensional By Frobenius theorem we can find local coordinates for X (x 1,, x k, x k+1,, x n ) = (x 1, x 2 ) such that D = span{ 1, 1,, 1 }

DISC Systems and Control Theory of Nonlinear Systems, 21 5 In such coordinates (i) implies that ẋ = f(x) + g(x)u takes the form Furthermore by (ii) we have ẋ 1 = f 1 (x 1, x 2 ) + g 1 (x 1, x 2 )u ẋ 2 = f 2 (x 2 ) + g 2 (x 2 )u L x i h(x) =, i = 1,, k, implying that h only depends on x 2, that is h(x 2 ) Finally by (iii) the disturbance vectorfields e j are in D

DISC Systems and Control Theory of Nonlinear Systems, 21 6 It follows that the system has the form ẋ1 = f1 (x 1, x 2 ) + g1 (x 1, x 2 ) ẋ 2 f 2 (x 2 ) g 2 (x 2 ) u + e1 (x 1, x 2 ) d y = h(x 2 ) and thus d only affects the x 1 -dynamics, and therefore does not affect y

DISC Systems and Control Theory of Nonlinear Systems, 21 7 Next thing: find conditions such that a distribution D satisfying conditions (ii) and (iii) can be rendered invariant by static state feedback Definition 2 A distribution D on X is called controlled invariant for the system ẋ = f(x) + g(x)u if there exists a regular state feedback u = α(x) + β(x)v, detβ(x), x X, such that, denoting the closed-loop system by ẋ = f(x) + m j=1 g j (x)v j f(x) = f(x) + g(x)α(x), g(x) = [g(x)β(x)]

DISC Systems and Control Theory of Nonlinear Systems, 21 8 we have [ f, D] D, [ g j, D] D, j = 1,, m Key observation: By the property [X γ, Y ] = γ[x, Y ] X L Y γ, this implies [f, D] D + G, [g j, D] D + G, j = 1,, m where G = span{g 1,, g m }

DISC Systems and Control Theory of Nonlinear Systems, 21 9 Theorem 3 Consider the system ẋ = f(x) + g(x)u on X, and let D be a constant dimensional and involutive distribution Suppose that (i) dim[d(x) + G(x)] = constant (ii) [f, D] D + G (iii) [g j, D] D + G, j = 1,, m Then for every point x X there exists a neighborhood V of x and a regular feedback such that [ f, D] D, [ g j, D] D, j = 1,, m The distribution D is therefore called locally controlled invariant

DISC Systems and Control Theory of Nonlinear Systems, 21 1 Proof Take local coordinates x = (x 1,, x k, x k+1,, x n ) = (x 1, x 2 ) around x such that D = span{ x 1,, } =: span{ x k x 1 } Write correspondingly f = f1 f 2, g = g1 g 2 By (i) the (n k, m)-matrix g 2 (x) has constant rank, say l Without loss of generality we may assume that the first l rows of g 2 (x) are independent

DISC Systems and Control Theory of Nonlinear Systems, 21 11 Lemma 4 There exists, locally around x, an invertible matrix β(x) such that for some R(x) g 2 (x)β(x) = I l R(x) l (m l) (n k l) (m l) n k m It follows that (unspecified elements denoted by ) g(x) := g(x)β(x) = I l R(x) } k } l } n k l

DISC Systems and Control Theory of Nonlinear Systems, 21 12 Then (iii) yields for i = 1,, k [ g, ] (x) = x i R x i (x) im I k + im I l R(x) which necessarily implies that R x i (x) =, i = 1,, k, and thus that [ g, ] (x) D(x), x i i = 1,, k or equivalently, [ g j, D] D, j = 1,, m Hence β(x) is as required

DISC Systems and Control Theory of Nonlinear Systems, 21 13 In order to construct α(x) write f 2 as f 2 = f21 f 22 Now define α(x) = β(x) f21 (x) l m l

DISC Systems and Control Theory of Nonlinear Systems, 21 14 Then f = f + gα = f 1 f 21 f 22 I l R f21 = f 22 Rf 21 and (ii) implies for i = 1,, k [ f, x i ] = x i (f 22 Rf 21 ) im I k + im I l R showing, as above, that equivalently x i (f 22 Rf 21 ) =, i = 1,, k, or [ f, D] D

DISC Systems and Control Theory of Nonlinear Systems, 21 15 Proof of Lemma Denote the matrix consisting of the first l independent rows of g 2 (x) by g 21 (x) Consider the equation g 21 (x)β(x) [I l l (m l) ] = Clearly in x this has an invertible solution β(x ) since g 21 (x ) has full row rank Then by the implicit function theorem it follows that locally around x there exists an invertible solution β(x) of g 2 (x)β(x) = I l R(x) l (m l) Since the rank of g 2 (x) is l, also the rank of g 2 (x)β(x) is l, and thus necessarily the unspecified elements have to be zero

DISC Systems and Control Theory of Nonlinear Systems, 21 16 Example 5 Consider a linear system ẋ = Ax + Bu and let D be the distribution corresponding to a linear subspace V R n, ie, if V = span{e 1,, e k }, then D = span{ x 1,, x k } Then (ii) amounts to AV V + imb Furthermore, (iii) is automatically satisfied since [b j, x i ] = for the constant columns b j of B

DISC Systems and Control Theory of Nonlinear Systems, 21 17 Theorem 6 Suppose there exists an involutive and constant dimensional distribution D on X such that (i) [f, D] D + G, [g j, D] D + G (ii) dim(d + G) is constant (iii) D kerdh (iv) e j D, j = 1,, l Then around each x X there exists a regular state feedback u = α(x) + β(x)v such that the closed-loop system is disturbance decoupled

DISC Systems and Control Theory of Nonlinear Systems, 21 18 Example 7 Consider the cart with fixed rear axis d dt x 1 x 2 ϕ θ = cos(ϕ + θ) sin(ϕ + θ) sinθ u 1 + 1 u 2 with u 1 the driving input, and u 2 the steering input

DISC Systems and Control Theory of Nonlinear Systems, 21 19 Suppose there is additionally a disturbance vector field e = ( 1 1) T (corresponding to sideways slipping of the rear axis) The distribution D spanned by e, ie, D = span{ 1 1 } is involutive and constant dimensional, and satisfies the conditions Define x 3 := ϕ + θ, x 4 := θ Then the full dynamics is d dt x 1 x 2 x 3 x 4 = cosx 3 sin x 3 sin x 4 u 1 + 1 1 u 2 + 1 d

DISC Systems and Control Theory of Nonlinear Systems, 21 2 while D = span{ } 1 Thus modulo permutations we have obtained the required coordinates, with g 2 (x) = cosx 3 sinx 3 sinx 4 1

DISC Systems and Control Theory of Nonlinear Systems, 21 21 Consider now an arbitrary point x = (x 1, x 2, x 3, x 4) Suppose cosx 3 Then the first and third row of g 2 (x) are independent, and we construct β(x) such that cosx 3 sinx 3 β(x) = 1 sinx 4 1 1 A possible solution is β(x) = 1 cos x 3 sin x 4 cos x 3 1

DISC Systems and Control Theory of Nonlinear Systems, 21 22 If cosx 3 =, then sinx 3, in which case the second and third row of g 2 (x) are independent, and we construct β(x) such that cosx 3 sinx 3 sinx 4 1 β(x) = 1 1 yielding as possible solution β(x) = 1 sin x 3 sin x 4 sin x 3 1

DISC Systems and Control Theory of Nonlinear Systems, 21 23 Both feedback expressions are only locally defined In the present case we may also find a globally defined β(x) which solves the disturbance decoupling problem: β(x) = 1 sinx 4 1

DISC Systems and Control Theory of Nonlinear Systems, 21 24 transforming the input vectorfields g 1 and g 2 to g 1 = cos(ϕ + θ) sin(ϕ + θ) sinθ, g 2 = 1 1 Note that the third entry of the new driving input vectorfield g 1 is zero, and thus ϕ + θ = implying that ϕ + θ remains constant Hence the front axis moves in the same direction Furthermore, since θ = sinθ the angle θ converges to zero, and therefore the rear axis tends to a position parallel to the front axis

DISC Systems and Control Theory of Nonlinear Systems, 21 25 Conclusion: solution of the disturbance decoupling problem has been reduced to the search for a distribution D satisfying all the conditions How do we find such a distribution? Answer Compute the maximal distribution satisfying conditions (i)-(iii): Theorem 8 Compute the maximal distribution D satisfying conditions (i)-(iii) Suppose D is constant dimensional and D + G is constant dimensional Then the disturbance decoupling problem is solvable around any x X if and only if e j D, j = 1,, l

DISC Systems and Control Theory of Nonlinear Systems, 21 26 Algorithm for computing D Define the sequence of distributions D µ, µ = 1, 2,, as D 1 = kerdh D µ+1 = kerdh span{x vectorfield [f, X] D µ + G, [g j, X] D µ + G, j = 1,, m} Then (i) D 1 D 2 D 3 (ii) D µ is involutive for µ = 1, 2, (iii) Denote D = lim µ Dµ If D satisfies (i), (iii) then D D (iv) D satisfies (i) and (iii)

DISC Systems and Control Theory of Nonlinear Systems, 21 27 Proof (i) Clearly D 1 D 2 Suppose D µ D µ+1 Then D µ+2 = kerdh {X vectorfield [f, X] D µ+1 + G [g j, X] D µ+1 + G, j = 1,, m} kerdh {X vector field [f, X] D µ + G [g j, X] D µ + G, j = 1,, m} = D µ+1 (ii) D 1 = kerdh is involutive Indeed, let X, Y kerdh, then L [X,Y ] h = L X (L Y h) L Y (L X h) =, and thus [X, Y ] kerdh Now

DISC Systems and Control Theory of Nonlinear Systems, 21 28 suppose D µ is involutive, and let X, Y be in D µ+1, ie [f, X] D µ + G, [g j, X] D µ + G, [f, Y ] D µ + G, [g j, X] D µ + G Then by the Jacobi-identity, it follows that for some Z X, Z Y D µ + G [f, [X, Y ]] = [[f, X], Y ] + [X, [f, Y ]] = = [Z X, Y ] + [X, Z Y ] D µ + G and similarly [g j, [X, Y ]] D µ + G, j = 1,, m

DISC Systems and Control Theory of Nonlinear Systems, 21 29 (iii) Take any D satisfying (i) and (iii), Then D D 1 = kerdh Suppose D D µ, then by (i) [f, D] D + G D µ + G and [g j, D] D + G D µ + G, j = 1,, m, and thus D D µ+1 It follows that D D µ, µ = 1, 2, (iv) By construction D = kerdh {X vector field [f, X] D + G, [g j, X] D + G}, from which everything follows

DISC Systems and Control Theory of Nonlinear Systems, 21 3 Special case Consider the single-input single-output system ẋ = f(x) + g(x)u y = h(x) Let ρ be the smallest nonnegative integer such that the function L g L ρ f h(x) is different from the zero-function Assume this function is nowhere equal to zero Then D = ker(spandh, dl f h,,dl ρ f h) Similar expressions will hold for the multi-input multi-output case whenever the input-output decoupling matrix has maximal rank everywhere; cf Chapter 8 and next lecture