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1 Definitions Xu Chen February 19, 2021 Outline • Preliminaries • Stability concepts • Equilibrium point • Stability around an equilibrium point • Stability of LTI systems • Method of eigenvalue/pole locations • Routh Hurwitz criteria for CT and DT systems • Lyapunov’s direct approach 1 Definitions 1.1 Finite dimensional vector norms Let v 2 Rn. A norm is: • a metric in vector space: a function that assigns a real-valued length to each vector in a vector space, e.g., p T p 2 2 2 – 2 (Euclidean) norm: kvk2 = v v = v1 + v2 + ··· + vn Pn – 1 norm: kvk1 = i=1 jvjj (absolute column sum) – infinity norm: kvk1 = maxi jvij Pn p 1=p – p norm: jjvjjp = ( i=1 jvij ) ; 1 ≤ p < 1 • default in this set of notes: k · k = k · k2 1.2 Equilibrium state For n-th order unforced system x_ = f (x; t) ; x(t0) = x0 an equilibrium state/point xe is one such that f (xe; t) = 0; 8t • the condition must be satisfied by all t ≥ 0. • if a system starts at equilibrium state, it stays there • e.g., (inverted) pendulum resting at the verticle direction • without loss of generality, we assume the origin is an equilibrium point 1.3 Equilibrium state of a linear system For a linear system x_(t) = A(t)x(t); x(t0) = x0 • origin xe = 0 is always an equilibrium state • when A(t) is nonsingular, multiple equilibrium states exist 1 Xu Chen 1.4 Continuous function February 19, 2021 Figure 1: Continuous functions Figure 2: Uniformly continuous (left) and continuous but not uniformly continuous (right) functions 1.4 Continuous function The function f : R ! R is continuous at x0 if 8 > 0, there exists a δ (x0; ) > 0 such that jx − x0j < δ =) jf (x) − f (x0)j < Graphically, continuous functions is a single unbroken curve: 2 1 x2−1 e.g., sin x, x , x−1 , x−1 , sign(x − 1:5) 1.5 Uniformly continuous function The function f : R ! R is uniformly continuous if 8 > 0, there exists a δ () > 0 such that jx − x0j < δ () =) jf (x) − f (x0)j < • δ is a function of not x0 but only 1.6 Lyapunov’s definition of stability • Lyapunov invented his stability theory in 1892 in Russia. Unfortunately, the elegant theory remained unknown to the West until approximately 1960. • The equilibrium state 0 of x_ = f(x; t) is stable in the sense of Lyapunov (s.i.L) if for all > 0, and t0, there exists δ (, t0) > 0 such that kx (t0) k2 < δ gives kx (t) k2 < for all t ≥ t0 2 Xu Chen 1.7 Uniform stability in the sense of Lyapunov February 19, 2021 Figure 3: Stable in the sense of Lyapunov: kx (t0) k < δ ) kx (t) k < 8t ≥ t0. Figure 4: Asymptotically stable in the sense of Lyapunov: kx (t0) k < δ ) kx (t) k ! 0. 1.7 Uniform stability in the sense of Lyapunov • The equilibrium state 0 of x_ = f(x; t) is uniformly stable in the sense of Lyapunov if for all > 0, there exists δ () > 0 such that kx (t0) k2 < δ gives kx (t) k2 < for all t ≥ t0 • δ is not a function of t0 1.8 Lyapunov’s definition of asymptotic stability The equilibrium state 0 of x_ = f(x; t) is asymptotically stable if • it is stable in the sense of Lyapunov, and • for all > 0 and t0, there exists δ (, t0) > 0 such that kx (t0) k2 < δ gives x (t) ! 0 1.9 Uniform and global asymptotic stability • the origin is uniformly and asymptotically stable if – it is asymptotically stable and – δ does not depend on time x – e.g. x_ = − 1+t is asymptotically stable but not uniformly stable • the origin is globaly and asymptotically stable if – it is asymptotically stable, and – δ can be made arbitraily large (it does not matter how far the initial condition is from the origin) 3 Xu Chen February 19, 2021 2 Stability of LTI systems: method of eigenvalue/pole locations The stability of the equilibrium point 0 for x_ = Ax or x(k + 1) = Ax(k) can be concluded immediatelly based on the eigenvalues, λ’s, of A: At λt λt σt σt • the response e x(t0) involves modes such as e , te , e cos !t, e sin !t k k k−1 k k • the response A x(k0) involves modes such as λ , kλ , r cos kθ, r sin kθ • eσt ! 0 if σ < 0; eλt ! 0 if λ < 0 p • λk ! 0 if jλj < 1; rk ! 0 if jrj = σ2 + !2 = jλj < 1 2.1 Stability of the origin for x_ = Ax stability of λi(A) x_ = Ax at 0 unstable Re fλig > 0 for some λi or Re fλig ≤ 0 for all λi’s but for a repeated λm on the imaginary axis with multiplicity m, nullity (A − λmI) < m (Jordan form) stable i.s.L Re fλig ≤ 0 for all λi’s and for any repeated λm on the imaginary axis with multiplicity m, nullity (A − λmI) = m (diagonal form) asymptotically Re fλig < 0 for all λi (such a matrix is called a Hurwitz matrix) stable Example 1. Unstable system 0 1 x_ = Ax; A = 0 0 0 1 λ = λ = 0 m = 2 (A − λ I) = = 1 < m • 1 2 , , nullity i nullity 0 0 • i.e., two repeated eigenvalues but needs a generalized eigenvector ) Jordan form after similarity transform 1 t eAt = t • verify by checking 0 1 : grows unbounded Example 2. Stable in the sense of Lyapunov 0 0 x_ = Ax; A = 0 0 0 0 λ = λ = 0 m = 2 (A − λ I) = = 2 = m • 1 2 , , nullity i nullity 0 0 1 0 eAt = • verify by checking 0 1 2.2 Routh-Hurwitz criterion • The asymptotic stability of the equilibrium point 0 for x_ = Ax can also be concluded based on the Routh- Hurwitz criterion. – The Routh Test (by E.J. Routh, in 1877) is a simple algebraic procedure to determine how many roots a given polynomial n n−1 A(s) = ans + an−1s + ··· + a1s + a0 has in the closed right-half complex plane, without the need to explicitly solve the characteristic equation. – German mathematician Adolf Hurwitz independently proposed in 1895 to approach the problem from a matrix perspective . – popular if stability is the only concern and no details on eigenvalues (e.g., speed of response) are needed • simply apply the Routh Test to A(s) = det (sI − A) −1 • Recap: The poles of transfer function G(s) = C (sI − A) B + D come from det (sI − A) in computing the −1 inverse (sI − A) . 4 Xu Chen 2.3 The Routh Array February 19, 2021 2.3 The Routh Array n n−1 For A(s) = ans + an−1s + ··· + a1s + a0, construct n s an an−2 an−4 an−6 ··· n−1 s an−1 an−3 an−5 an−7 ··· n−2 s qn−2 qn−4 qn−6 ··· n−3 s qn−3 qn−5 qn−7 ··· . 1 s x2 x0 0 s x0 • first two rows contain the coefficients of A(s) • third row of the Routh array is constructed from the previous two rows via · a b x · · c d y · bc − ad xc − ay · · c c · ···· • All roots of A(s) are on the left half s-plane if and only if all elements of the first column of the Routh array are positive. • Special cases: – If the 1st element in any one row of Routh’s array is zero, one can replace the zero with a small number and proceed further. – If the elements in one row of Routh’s array are all zero, then the equation has at least one pair of real roots with equal magnitude but opposite signs, and/or the equation has one or more pairs of imaginary roots, and/or the equation has pairs of complex-conjugate roots forming symmetry about the origin of the s-plane. – There are other possible complications, which we will not pursue further. See, e.g. "Automatic Control Systems", by Kuo, 7th ed., pp. 339-340. Example 3. A(s) = 2s4 + s3 + 3s2 + 5s + 10 s4 2 3 10 s3 1 5 0 2 2×5 s 3 − 1 = −7 10 0 1 1×10 s 5 − −7 0 0 s0 10 0 0 • two sign changes in the first column • unstable and two roots in the right half side of s-plane 2.4 Stability of the origin for x(k + 1) = f (x(k); k) • the theory of Lyapunov follows analogously for nonlinear time-varying discrete-times systems of the form x (k + 1) = f (x(k); k) ; x (k0) = x0 • equilibrium point xe: f (xe; k) = xe; 8k • without loss of generality, 0 is assumed an equilibrium point. 5 Xu Chen 2.5 Stability of the origin for x(k + 1) = Ax(k) February 19, 2021 2.5 Stability of the origin for x(k + 1) = Ax(k) • 0 is always an equilibrium point, althoughly not necessarily the only one stability of λi(A) x(k + 1) = Ax(k) at 0 unstable jλij > 1 for some λi or jλij ≤ 1 for all λi’s but for a repeated λm on the unit circle with multiplicity m, nullity (A − λmI) < m (Jordan form) stable i.s.L jλij ≤ 1 for all λi’s but for any repeated λm on the unit circle with multiplicity m, nullity (A − λmI) = m (diagonal form) asymptotically jλij < 1 for all λi (such a matrix is called a Schur matrix) stable 2.6 Routh-Hurwitz criterion for discrete-time LTI systems • The stability domain jλij < 1 for discrete-time systems is a unit disk.
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