Robust Control 2 Controllability, Observability & Transfer Functions Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /26/24
Outline Reachable Controllability Distinguishable Observability Zeros Transfer Functions Poles Interconnections Stability
System x = Ax+ Bu y= Cx x R, u R, y R n m p x(t; x,u) e x e = + At t A(t-s) Bu(s)ds
Reachability Definition: A state x R is reachable from x if there n exists a finite time t > and a piecewise continuous control ( ) u such that x(t;x,u) = x. R x denotes the set of states reachable from x.
Reachability: Properties If x is reachable from x in some time t >, it is reachable in every time t. To see this simply rescale s: t t A(t-s) A(t st - t ) = st t st e Bu(s)ds e Bu( )d = t ( t ) ( t ) A(t- st t ) A(t-t) st st t e Be u( )d A(t -t) Thus, we have the replacement us ( ) e u( ). Notice that x is reachable from x if and only if x e x At is reachable from the origin for any < t < t t At A(t-s) At A(t-s) = + e x = e Bu(s)ds x e x e Bu(s)ds x st t
Some Geometry, Consider two linear vector spaces X, X with inner products 2 2 { y X y = Ax, x X } 2 2, and,,respectively, and a mapping A: X X - the range or image ( Im ) of A is the set of points in X - the null space or kernel ( ker ) of A is the set of points in X { x X Ax= } * - the adjoint mapping A : X2 X is defined by yax *, = Ayx, 2 2
Some Geometry, 2 * If then is called self-adjoint A= A A ( A) * * * A = A A = A It is always true that and * * If then is called normal AA= AA A ( ) The following decompositions of finite dimensional linear vector spaces are true: X 2 = ker Im ( * A ) ( * ) Im ( ) X = ker A A These are orthogonal decompositions, i.e., ( ) ( * ) x ker A, y Im A x, y =
Some Geometry, 3 x 3 y = Ax, A= ker A y 2 x 2 y x Im A T T T x = A y, A =
Reachability Condition Let U denote the linear vector space of control functions u( τ ), τ [, t ], ( ) ( ) n and X R the space of states x t. The map A : U X is defined by At ( τ ) xt ( ) = e Bu( τ) dτ x * * R x ImA x ImA note : Proposition: t [ t ] the time interval, is T τ T At ( ) A ( t τ) e BB e d A AA X X The set of states reachable from the origin over * ImAA = Im t τ
Controllability Definition: The system or the matrix pair ( AB, ) is said to be (completely) controllable if any state x is reachable from any other state x in finite time. Proposition: only if where The system is completely controllable if and rank G ( t ) = n C T At ( τ) A ( t ) G ( t ) e BB T τ e d C = is the controllability Grammian. t τ
Controllability Main Result B : = Im( B) B : = B+ B+ + B= Im n n A A A BAB A B Theorem: R ( ) = A B If the system is completely controllable there is a unique control T T A ( t t () ) ( ) ut = Be G t x C that steers the origin to x in precisely time t.
Distinguishable Definition: A state x R is indistinguishable from x n if for every finite time t and piecewise continuous control ( ) ut ( ), ytx ( ;, u) = ytx ( ;, u). I x denotes the set of states 2 indistinguishable from x. C n N : ( i CA = ker CA ) = ker i= CA I() = N Theorem: n
Observability Definition: The system or the matrix pair ( C, A) is said to be (completely) observable if knowledge of ut ( ) and yt ( ) on a finite time interval determines the state trajectory on that interval. Theorem: The system or the matrix pair ( C, A) is (completely) observable if and only if I () C CA rank = n n CA =, i.e.
Summary: Controllability/Observability Controllability rank = C CA Observability rank = n CA Kalman Decomposition, x z such that d dt n B AB A B n z A A2 A3 A4 z B z z 2 A22 A 24 z 2 B 2 z = 2 + u, y = [ C 2 C ] 4 z 3 A33 A 34 z 3 z 3 z4 A44 z4 z 4 z, z controllable z, z observable 2 2 4 Notice that the substate z 2 is both controllable and observable
Example 2 2 x = x u, y [ ] x 4 + = 4 C = [ B AB] = 4 s + 4 2 s + 2 G s = C si A B= + + ( ) [ ] [ ] ( s 2 )( s 4 ) ( s + 2) = = ( s+ 2)( s+ 4) s+ 4
Some MATLAB Functions Function canon ctrb ctrbf gram obsv obsvf ss2ss ssbal minreal canonical stste-space realizations controllability matrix controllability staircase form controllability and observability gramians observability matrix observability staircase form state coordinate transformation diagonal balancing of state-space realizations returns a minimal realization
Example, Continued >> A=[-2-2;,-4]; >> B=[;]; >> C=[ ]; >> sys=ss(a,b,c,); >> ctrb(sys) ans = -4-4 >> gram(sys,'c') ans =.25.25.25.25 >> tf(sys) Transfer function: s + 2 ------------- s^2 + 6 s + 8 >> tf(minreal(sys)) state removed. Transfer function: ----- s + 4
System Poles & Zeros Two descriptions of linear time-invariant systems state space and transfer function. x = Ax+ Bu y = Cx+ Du ( ) [ ] G s = C si A B+ D Assumption: G is a complete characterization of A, B, C, D or, equivalently, ABCD,,, is a minimal realization of G. Defining poles via state space is very easy: the poles of G are the eigenvalues of A. Defining zeros is more complicated. We do it via state space in the followin g.
SISO System Zeros recall: x = Ax+ Bu n x R, u R, y R y = Cx+ Du { } [ ] [ ] Y() s = C si A x + C si A B+ D U() s suppose: λt () { [ ] } ns () u t = e, G( s): = C si A B + D = k, d ( s) = si A ds () [ ] [ ] = + k = + + ds () ds () s λ ds () ds () s λ C si A x ns () C si A x ns () G( λ) (), Y s
SISO System Zeros, Cont d x can always be chosen so that ( ) [ ] C si A x ns () + ds () d() s in which case ( λ ) G Y() s =, if λ is a zero of G( λ), then Y ( s) =, y( t) = s λ In summary: if λ is a system zero, there exists x such that x t = x λt and ut ( ) = e y t ()
MIMO System Zeros x = Ax+ Bu y = Cx+ Du x R, u R, y R n m p m n Does there exist g R and x R such that λt λt ut () = ge xt () = xe and yt ()? The assumed solution must satisfy λt λt λt λxe = Axe + Bge λi A B x = Cx e Dge C D g λt λt = +
MIMO System Zeros, Cont d This represents n+ p equations in n+ m unknowns. Suppose λi A B r = rank C D r = n+ min( m, p) max r< n+ min( m, p) nontrivial sol'ns p< m always nontrivial sol'ns r= r = n+ p m p max max independent sol'ns p> m and r= r = n+ m there are no nontrivial sol'ns
Square MIMO Systems (p=m) Nondegenerate case: If for typical λ, λi A B r = rank n m C D = + Those specific values of λ for which r < n+ m are called invariant zeros. Invariant zeros consist of input decoupling zeros (uncontrollable modes), λ satisfies rank [ λ ] I A B < n output decoupling zeros (unobservable modes), λ satisfies λi A rank n C < transmission zeros, all other invariant zeros.
Square MIMO Systems, Cont d degenerate case: ( λ ) For typical λ, λi A B r = rank < n+ m C D G insufficient independent controls,rank B< m insufficient independent outputs, rank C < p
Transfer Functions x = Ax+ Bu y = Cx ( ) = [ ] = [ ] 2 2 2 x R, u R, y R n m p G s C si A B C si A B 2 2 2 where A, B, C are those of the Kalman decomposition -, i.e., parameters of a minimal realization. So, only the controllable and observable part of the system is characterized by its transfer function. ( ) Definition: G s is called a complete characterization of the system if the system is completely observable and controllabl e.
Poles & Zeros from Transfer Functions /26/24
Numbers: Prime & Coprime A prime number (or integer) is a positive integer p > that has no positive integer divisors other than and itself. Two integers are relatively prime or coprime if they share no positive integer factors (divisors) other than - i.e., their greatest common divisor is. Bezout's identity: exist integers x and y such that (, ) GCD a b = ax + by If a and b are integers not both zero, then there If a and b are coprime then there exist integers x and y such that = ax + by
Polynomials These ideas have been extended to polynomials, matrices with polynomial elements matrices with rational elements Two polynomials ( ) ( ) n s = a s + a s + + a, a m m m m m d s = b s + b s + + b, b n n n n are coprime if their greatest common divisor is a nonzero constant, i.e., they have no common factors.
Example F = s 4 + 2s 3 + s + 2 = (s + ) (s + 2) (s 2 - s + ) F 2 = s 5 + s 4 + 2s 3 + 3s 2 + 3s + 2 = (s + ) (s 2 - s + 2) (s 2 + s + ) gcd(f,f 2 ) = (s + ) Bezout relation: (5/24s 3 + /2s 2 + /4s + 5/24) F + (-5/24s 2-7/24s + 7/24) F 2 = s +
Polynomial Matrices Two matrix polynomials m m ( ) = + + + n i ( ) = n + n + + right common divisor R( s) R ( ) = ( ) ( ), ( ) = ( ) ( ) ( ) D( s) right coprime R( s) N s A s A s A m m pa, q pb, q A R, Bi R D s B s B s B qq, have a if N s N s R s D s D s R s N s, are if the only right common divisors are unimodular, i.e., det = c. Similary, left coprimeness can be defined for polynomial matrices with the same number of rows.
Poles ( ) Suppose the q m transfer matrix G s is a complete characterization of x = Ax+ Bu, y = Cx+ Du ( ) can always be factored into G( s) = D ( s) N ( s) = N ( s) D ( s) G s l l r r where D, N and D, N are coprime pairs of polynomial matrices. N l l r r and D, D are called numerator, denominator matrices, respectively. Theorem: [ si A] = D s = D ( s) det α det ( ) α det α, α are constants. Definition: Poles are the roots of: det r r l 2 r 2 [ si A], or det D ( s), or det D ( s) = = = l r r, N l
Poles & Zeros from Transfer Functions Assume G(s) is a complete characterization. Theorem: The pole polynomial (s) is the least common denominator of all non-identically-zero minors of all orders of G(s). Theorem: The zero polynomial is the greatest common divisor of all numerators of all order-r minors of G(s), where r is the generic rank of G(s), provided that these minors have been adjusted to have (s) as there denominator.
Example Recall a minor of a matrix is the determinant of a matrix obtained by deleting rows and columns. Consider the transfer function: s 4 G( s) = 2 4.5 2( s s + ) To determine poles we need all minors of all orders. The 4 minors of order are s 4 4.5 s,,,2 s+ 2 s+ 2 s+ 2 s+ 2 ( ) rank G s =2. ( ) The single minor of order 2 is det G s = 2 ( s) ( ) pole polynomial: φ = s+ 2 zero polynomial: z s = s 4 s 4 ( s + 2)
Multivariable Interconnections & Feedback Loops /26/24
Well-Posed Loops: Example G( s) s G( s) = s 2 s+ s+ s+ s Gcl ( s) = G( s) I + G( s) = s Theorem: Let GH, be proper rational transfer matrices. Then cl [ ] G = G I + HG ( ) ( ) is proper and rational iff I + H G is nonsingular.
Poles of Closed Loops G( s) H ( s) cl [ ] G = G I + HG It might be anticipated that the poles of G are the roots of det [ I + HG] s s s s+ G( s) =, H( s) = I2 2 s + 2s s s+ s+ Gcl ( s) =, det [ I + HG] s 2 s obviously, has poles at s =±.. Not True!! cl
Poles, Cont d Theore m: ( ) If GH, are proper, reational matrices and ( ) ( ) det I + H G, then the poles of Gcl are the roots of the polynomial G s ( s) ( s) ( s) det I H( s) G( s) = G H + Example: + = s +.5 ( s.5)( s.5) ( s. 5) ( s.5)( s+.5 ) ( s.5), H = I 2
Poles: Example Cont d s +.5 det I + G( s) = s.5 = + = = + + G G cl ( s) ( s.5)( s.5 ), ( s) ( s) ( s.5)( s.5) ( s) ( s.5) = ( s+.5) ( s+.5) s +.5 = ( s+.5)( s+.5) s+.5 s+.5 H
System Interconnections G Systems G, G are complete characterizations, with 2 G = D N = N D, i =, 2 coprime fractions. i li li ri ri parallel connection G G 2 G 2 controllable D, D left coprime r r2 observable D, D right coprime series connection l l2 G controllable D, N, or D D, N or D, N N are left coprime r2 r l r2 l l2 l2 r G 2 observable D, N, or D D, N or D, N N right coprime ( ) ( ) det I + G2 G. Then controllable GG controllable l r2 l r2 r2 r l2 r 2 observable G 2 G observable
Stability x = Ax+ Bu y = Cx+ Du ( ) [ ] x R, u R, y R G s = C si A B+ D n m p The basic idea is that stable system responds to a perturbation by remaining within a neighborhood of its equilibrium point. a state perturbation with zero input (Lyapunov/Asymptotic) an input perturbation with zero state (BIBO) simultaneous state and input perturbation (Total) ( A) Lyapunov: Reλ, eigenvalues with Reλ = have full set of eigenvectors. BIBO: ( A) ( ) Asymptotic: Reλ < poles of G s < ( ) Total: Lyapunov + poles of G s <