# Solution of Linear State-space Systems

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1 Solution of Linear State-space Systems Homogeneous (u=0) LTV systems first Theorem (Peano-Baker series) The unique solution to x(t) = (t, )x 0 where The matrix function is given by is called the state transition function It satisfies the following properties 1. For every t and every t0, (t, ) is the unique solution of (t, )=I + d dt ẋ(t) =A(t)x(t), x( )=x 0 2 R n + A(s 1 ) ds 1 + (t, ) applez s1 A(s 1 ) applez s1 applez s2 (t, )=A(t) (t, ), (, )=I A(s 2 ) ds 2 ds 1 A(s 1 ) A(s 2 ) A(s 3 ) ds 3 ds 2 ds proof by substitution 2. For any s, (t, )= (t, s) (s, ) [the semigroup property] 3. For every t and t0, (t, ) is nonsingular and since I = (,t) (t, )= (t, ) (,t) 1

2 Solution to Nonhomogeneous LTV Systems Theorem (Variation of constants) The unique solution to is ẋ(t) =A(t)x(t)+B(t)u(t), y(t) =C(t)x(t)+D(t)u(t), x( )=x 0 x(t) = (t, )x 0 + Proof by substitution again Uniqueness follows by the global Lipschitz property of the mapping which is due to linearity The zero-input response is y(t) =C(t) (t, )x 0 + (t, )B( ) d C(t) (t, )B( ) d + D(t)u(t) y zero-input (t) =C(t) (t, )x 0 The zero-state response is y zero-state (t) = C(t) (t, )B( ) d + D(t)u(t) 2

3 The unique solution of is given by Discrete-time LTV Systems where the discrete transition function, Properties of 1. (t, ) is the unique solution of 2. For every t s, (t, )= (t, s) (s, ) There is no requirement for (t, ) to be nonsingular will be singular if any A(s),, is singular The unique solution of is (t, ) x(t + 1) = A(t)x(t), x( )=x 0, t x(t) = (t, )x 0, t (t, ), is given by ( I, t = (t, )= A(t 1)A(t 2)...A( + 1)A( ), t > x(t) = (t, )x 0 + (t +1, )=A(t) (t, ), (, )=I (t, ) t>s Xt 1 y(t) =C(t) (t, )x 0 + x(t + 1) = A(t)x(t)+B(t)u(t), x( )=x 0 y(t) =C(t)x(t)+D(t)u(t) = (t, + 1)B( )u( ), t Xt 1 = C(t) (t, + 1)B( )u( )+D(t)u(t), t, t, 2 Z 3

4 The Matrix Exponential Solution of LTI State-space Systems Define for M 2 R n n its exponential e M = also in From the Peano-Baker series for the ODE we see that the transition function for the LTI system (t, ) = I + A ds 1 + A 2 (t, t 0 )=I + A(s 1 ) ds 1 + A(s 1 ) applez s1 + A(s 1 ) A(s 2 ) applez s1 applez s1 applez s2 ds 2 ds 1 + A 3 x( )=x 0 2 R n A(s 2 ) ds 2 ds 1 applez s1 applez s2 (I M) 1 = I + M + M 2 + M = I + A(t )+ 1 2 A2 (t ) A3 (t ) 3 + = e A(t ) 1X k=0 1 k M k R n n A(s 3 ) ds 3 ds 2 ds is ds 3 ds 2 ds The solution to is x(t) =e A(t ) x 0 + y(t) =Ce A(t ) x 0 + e A(t Ce A(t ) Bu( ) d ) Bu( ) d 4

5 Properties of the Matrix Exponential 1. The function is the unique solution of 2. For every t, 2 R we have but in general 3. For every, e At is nonsingular and (unless A and B commute) 4. Using the Cayley-Hamilton theorem, for every A 2 R n n, there exist n scalar functions { i (t) :i =0,...,n 1} such that 5. For every, 6. so e At if e At e A = e A(t+ ) e At e Bt 6= e (A+B)t t 2 R e At 1 = e At e At = nx 1 i (t)a i, 8t 2 R i=0 A 2 R n n Ae At = e At A e At = L 1 (si A) 1 = L 1 1 [adj(si A)]T det(si A) = L 1 1 (s 1) m 1 (s 2) m 2...(s k) m [matrix of polynomials in s] k = [R 11 ]e 1t +[R 12 ]te 1t + +[R 1m1 ]t m 1 1 e 1 t +[R 21 ]e 2t +[R 22 ]te 2t + +[R 2m2 ]t m 2 1 e 2 t + +[R k1 ]e kt +[R k2 ]te kt + +[R kmk ]t m k 1 e k t e At 0 as t 1 Re [ i(a)] < 0, i=1,...,n d dt eat = Ae At, e A0 = I 5

6 7. Since for any invertible T More on Matrix Exponentials 8. Using a generalized eigenvector matrix X, X 1 AX = J (Jordan form) (T AT 1 ) k = T AT 1 T AT 1 T AT 1 = TA k T 1 1X e TAT 1t 1 1X = k (T AT 1 ) k t k 1 = T k Ak t k T 1 = Te At T 1 since J is block diagonal and upper triangular, so is This yields the same eigenvalue relationship as earlier 9. Discrete-time equivalent Using the formal power series Z 1 (zi A) 1 = Z 1 z 1 (I z 1 A) 1 = Z 1 z 1 I + z 2 A + z 3 A = Z 1 1 [adj(zi A)]T det(zi A) we have k=0 e At = Xe Jt X 1 k=0 e Jt (I M) 1 = I + M + M 2 + M A k 0 as t 1if i(a) < 1, i=1,...n 6

7 Lyapunov Stability Definition An equilibrium, x*, of a system is stable in the sense of Lyapunov if given any > 0 there exists a ( ) > 0 such that for all x0 satisfying x x 0 < we have x x(t) < for all t>0 This is a definition applicable to all systems For linear systems it simplifies greatly x*=0 is the candidate equilibrium Lyapunov stability is a property of the state equation only Definition (Lyapunov stability for linear systems) ẋ(t) =A(t)x(t)+B(t)u(t), y(t) =C(t)x(t)+D(t)u(t) (i) is (marginally) stable in the sense of Lyapunov if for every the homogeneous state response x(t) = (t, )x 0, t is uniformly bounded (ii) is asymptotically stable in the sense of Lyapunov if, in addition, for every initial condition x( )=x 0 2 R n we have x(t) 0 as t 1 (iii) is exponentially stable if, in addition, there exist constant c, λ>0 such that for every initial condition x( )=x 0 2 R n we have kx(t)k applece (t ) kx 0 k for all t (iv) is unstable if it is not marginally stable in the sense of Lyapunov Lyapunov stability only treats the homogeneous system x( )=x 0 2 R n ẋ(t) =A(t)x(t) 7

8 Theorem (8.2) Lyapunov (internal) stability The H-CLTI (homogeneous continuous-time linear time-invariant system) Proof (i) is marginally stable if and only if all the eigenvalues of A have negative or zero real parts and al Jordan blocks corresponding to eignevalues with zero real parts are 1x1 (ii) is asymptotically stable if and only if all eigenvalues of A have strictly negative real parts (iii) is exponentially stable if and only if all eigenvalues of A have strictly negative real pats (iv) is unstable if and only if at least one eigenvalue of A has positive real part or has zero real part but the corresponding Jordan block is larger than 1x1 Follows from the properties of ẋ(t) =Ax(t), x 2 R n For LTI systems, asymptotic and exponential stability coincide e At This is NOT so for LTV systems - see any discussion of the Mathieu Equation ÿ(t)+[ + cos t]y(t) = 0 8

9 Stability of the Mathieu equation The Mathieu equation - an LTV system ÿ(t)+[ + cos t]y(t) = 0 9

10 Lyapunov Stability in General (Khalil) Nonlinear, time-varying, homogeneous, autonomous, continuous-time system where is locally Lipschitz from domain into The equilibrium x 2 D, where f(x )=0, is (i) stable if for every ε>0 there is a δ(ε)>0 such that (ii) unstable if it is not stable ẋ(t) =f(x) f : D R n D R n R n kx(0) x k < =) kx(t) x k <, 8 (iii) asymptotically stable if it is stable and δ can be chosen so that kx(0) x k < =) lim kx(t) x k =0 t1 (iv) globally asymptotically stable if it is asymptotically stable and δ may be chosen arbitrarily For nonautonomous systems introduce the ideas of uniform stability ẋ(t) =f(x, t) we have similar ideas, but we must Note that for linear systems any kind of stability is necessarily a global property 10

11 Lyapunov s theorem Let x=0 be an equilibrium of Let then x=0 is stable Moreover, if Lyapunov continued in the domain D be a continuously differentiable function such that then x=0 is asymptotically stable If D = R n and kxk 1 =) V (x) 1(radially unbounded) then x=0 is globally asymptotically stable V(x) is a scalar function of the n-vector x V : D R ẋ(t) =f(x) V (0) = 0 and V (x) > 0 in D {0} (positive definite) V (x) < 0 in D {0} V (x) apple 0 in D (decrescent) It represents something akin to energy for a dissipative system Sometimes it is easier to find V than it is to prove stability directly There are many other uses for Lyapunov functions e.g. domain of attraction estimation 11

12 Lyapunov Stability in Linear Systems Consider the continuous-time LTI system The following statements are equivalent (i) The system is asymptotically stable (ii) The system is globally asymptotically stable (iii) The system is globally exponentially stable (iv) All the eigenvalues of A have negative real parts (v) For every symmetric positive definite matrix Q there exists a unique solution to the (continuous-time) Lyapunov Equation A T P + PA = Q Moreover, P is symmetric and positive definite (vi) There exists a symmetric positive definite matrix P for which the Lyapunov matrix inequality holds A T P + PA < 0 If A is asymptotically stable the unique solution of the Lyapunov Equation is P = Z 1 ẋ = Ax V(x)=x T Px is a Lyapunov function for this system 0 e AT t Qe At dt 12

13 Discrete-time Linear Lyapunov Stability The discrete-time LTV system (i) marginally stable in the sense of Lyapunov if for every initial condition x( )=x 0 the homogeneous state response x(t) = (t, )x 0 is uniformly (in t and t0) bounded (ii) asymptotically stable in the sense of Lyapunov if, in addition, we have for every initial condition x(t) 0 as t 1 (iii) exponentially stable if there exist constants c, λ such that for every initial condition kx(t)k applec (t t0) kx 0 k, 8t (iv) unstable if it is not marginally stable in the sense of Lyapunov x(t + 1) = A(t)x(t) The homogeneous discrete-time LTI system x(t + 1) = Ax(t) is (i) marginally stable if and only if all the eigenvalues of A have magnitude less than or equal to one and all the Jordan blocks corresponding to eigenvalues equal to one have size 1x1 (ii) asymptotically and exponentially stable if and only if all of the eigenvalues of A have magnitude less than one (iii) unstable if and only if at least one eigenvalue of A has magnitude larger than one or magnitude equal to one and Jordan block larger than 1x1 is 13

14 Discrete-time Linear Stability For the discrete-time homogeneous LTI system following statements are equivalent (i) The system is asymptotically stable (ii) The system is exponentially stable (iii) All eigenvalues of A have magnitude less than one (iv) For every symmetric positive definite matrix Q there exists a unique solution P to the Stein Equation (Discrete-time Lyapunov Equation) Moreover, P is symmetric and positive definite (v) There exists a symmetric positive definite matrix P for which the following Lyapunov matrix inequality holds A T PA P<0 For stable A, use the solution A T PA P = Q P = 1X A jt QA j V(x)=x T Px is a discrete-time Lyapunov function for this system j=0 x(t + 1) = Ax(t) the 14

15 Discrete-time Lyapunov Stability Consider the discrete-time nonlinear homogeneous autonomous system Let x=0 be an equilibrium in D, as before Let then x=0 is stable Moreover, if asymptotically stable If V : D R x(t + 1) = f(x(t)) be a function such that along all solutions of the system V (0) = 0 and V (x) > 0 in D {0} (positive definite) V (x(t + 1)) = V (f(x(t))) apple V (x(t)) in D (decrescent) D = R n V (x(t)) = V (f(x(t))) V (x(t)) < 0 in D {0} and V(x) is radially unbounded then x=0 is g.a.s. Very similar ideas to continuous time Uses the Monotone Convergence Theorem then x=0 is 15

16 Discrete-time Lyapunov (Stein) Equation - saying more Consider the discrete-time LTI system Take as our candidate Lyapunov function V(x)=x T Px with P>0 satisfies positive definite and radially unbounded properties Suppose A T PA P = C T C where C is mxn so that the right-hand side of the discrete Lyapunov equation is only nonpositive definite Then But x(t + 1) = Ax(t) V [x(t + 1)] V [x(t)] = x(t + 1) T Px(t + 1) x(t) T Px(t) = x(t) T A T PA P x(t) = x(t) T C T Cx(t) =[Cx(t)] T [Cx(t)] apple 0 V [x(t + n)] V [x(t)] = V [x(t + n)] V [x(t + n 1)] + V [x(t + n 1)] V [(x(t + n 2)] + + V [x(t + 1)] V [x(t)] = x(t + n 1) T C T Cx(t + n 1) x(t + n 2) T C T Cx(t + n 2) x(t) T C T h Cx(t) i = x(t) T A (n 1)T C T CA (n 1) + A (n 2)T C T CA (n 2) + + C T C x(t) = x(t) T [O T O]x(t)] < 0, if [C, A] is observable 16

17 A Grown-ups Version of the Lyapunov Equation The continuous-time LTI system ẋ = Ax is g.a.s. if there exists a positive definite solution P to the Lyapunov matrix equation with the matrix pair [A,C] observable The discrete-time LTI system x(t + 1) = Ax(t) is g.a.s. if there exists a positive definite solution to the discrete-time Lyapunov matrix equation with the matrix pair [A,C] observable We see observability arise again A T P + PA = C T C A T PA P = C T C It has to do with the only solutions to the equation which yield zero derivative (difference) over a time interval are those which are zero V is positive definite and decreasing, so it converges So ΔV converges to zero and this can only happen if x 0 17

18 The response of this system is Input-Output Stability where Lyapunov deals with the first term provided C(t) is uniformly bounded Bounded-Input Bounded-Output (BIBO) stability deals with the second term Definition: The continuous-time LTV system is uniformly BIBO stable if there exists a constant g such that for every input u(t) its zero-state response satisfies sup ky zero-state (t)k appleg ku(t)k ẋ(t) =A(t)x(t)+B(t)u(t), y(t) =C(t)x(t)+D(t)u(t), x( )=x 0 y zero-input (t) =C(t) (t, )x 0 y zero-state (t) = t2[,1) y(t) =y zero-input (t)+y zero-state (t) C(t) (t, )B( ) d + D(t)u(t) sup t2[,1) Note that these are just (euclidean) vector norms not function norms The LTV system is BIBO stable if and only if every element of D(t) is uniformly bounded and sup t2[,1) [C(t) (t, )B( )] ij d < 1 18

19 The system Continuous-time LTI BIBO Stability ẋ = Ax + Bu, y = Cx + Du Z 1 0 is BIBO stable if and only if Ce A B ij d < 1 This condition is met if and only if every element of the transfer function matrix G(s) =D + C(sI A) 1 B only has poles with negative real part When the system is exponentially stable then it must also be BIBO stable The reason for the rather peculiar condition on the transfer function is that we need to be concerned with unstable pole-zero cancellations We saw some examples of systems with A matrices which were unstable while the transfer function was stable G(s) = A = , B = , C = i(a) ={ 3, 2, 1}, Lyapunov unstable s 2 1 s 3 +4s 2 + s 6 = (s + 1)(s 1) (s + 2)(s + 3)(s 1) = s +1 (s + 2)(s + 3) BIBO stable 19

20 Zero-state response Discrete-time Input-output Stability The discrete-time LTV system is uniformly BIBO stable if there exists a constant g such that sup ky zs (t)k appleg sup ku(t)k The system is uniformly BIBO stable if and only if For the discrete-time LTI system the following statements are equivalent (i) the system is uniformly BIBO stable (ii) sup t2n 1X =0 x(t + 1) = A(t)x(t) + B(t)u(t), y zs (t) = Xt 1 =0 =0 y(t) = C(t)x(t) + D(t)u(t) Xt 1 C(t) (t, + 1)B( )u( )+ D(t)u(t), 8 t2n (iii) all the poles of all the elements of the transfer function matrix lie strictly inside the unit circle. t2n [C(t) (t, )B( )] ij < 1, for i =1,..., m, j =1,...,k x + = Ax + Bu, y = Cx + Du [CA B] ij < 1, for i =1,..., m, j =1,...,k 20

21 Summary The ideas of LTV systems specialize in the case of LTI systems Likewise, many ideas from linear systems give guidance how to proceed with nonlinear systems Lyapunov stability is one example The concepts of an energy-like function and of dissipation are entirely generalizable Finding a Lyapunov function for a nonlinear system is often much simpler than proving stability by any other means The Lyapunov function can help find the region of attraction of an equilibrium Input-output stability differs dramatically from Lyapunov stability Ideas from normed function spaces come to the fore MAE281A, 281B begin to get a handle on all of this That is why we spent more time on these ideas 21

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