Nonlinear System

ME 583 Review. Based on Nonlinear Systems (Hassan K. Khalil) book.


Introduction

Dynamical system can be represented by a finite number of coupled ODEs: \[ \displaylines{\dot{x} = f(t, x, u) \\ y = h(t, x, u) } \] When \(f\) does not depend explicitly on \(u\), the state equation becomes: \[ \displaylines{\dot{x} = f(t, x) } \] Furthermore, the system is said to be autonomous or time invariant if \(f\) does not depend explicitly on \(t\): \[ \displaylines{\dot{x} = f(x) } \]

Compared to Linear systems, nonlinear systems have some unique phenomenons:

  1. Finite escape time:
    • Linear system: state can only go to infinity in infinite time.
    • Nonlinear system: can go in finite time.
  2. Multiple isolated equilibriums:
    • Linear system: can have only one isolated equi. point.
    • Nonlinear system: state may converge to one of several steady-state operating points, depending on the initial state of the system.
  3. Limit cycles:
    • Linear system: must have a pair of eigenvalues on the imaginary axis to oscillate, which is nonrobust (unstable to perturbations).
    • Nonlinear system: can go into an oscillation of fixed amplitude and frequency, irrespective of the initial state.
  4. Subharmonic, harmonic, or almost-periodic oscillations:
    • Linear system: produces an output of the same frequency under a periodic input.
    • Nonlinear system: can oscillate with frequencies that are submultiples or multiples. It may even generate an almost-periodic oscillation.
  5. Chaos:
    • Linear system: deterministic steady-state behavior.
    • Nonlinear system: can have more complicated steady-state behavior that is not equilibrium, periodic oscillation, or almost-periodic oscillation.
  6. Multiple modes of behavior: Nonlinear system may exhibit multiple models of behavior based on type of excitations. When the property of excitation change smoothly, the behavior mode can have discontinuous jump.

Second-Order System

Consider a second-order autonomous system: \[ \begin{equation}\displaylines{ \begin{split} \dot{x}_1 = f_1(x_1, x_2) \\ \dot{x}_2 = f_2(x_1, x_2) \end{split} } \label{second} \end{equation} \]

Qualitative Behavior of Linear Systems

For a second-order LTI system, \(\eqref{second}\) becomes: \[ \displaylines{\dot{x} = Ax } \] and the solution given \(x(0) = x_0\) is \[ \displaylines{x(t) = M e^{J t} M^{-1} x_0 } \] where \(J\) is the Jordan form of \(A\) and \(M\) is a real nonsingular matrix s.t. \(M^{-1} A M = J\). \(J\) can have three forms depending on the eigenvalues of \(A\).

Case 1

\(\lambda_1 \ne \lambda_2 \ne 0\), \(J = \begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix}\)

The change of coordinates \(z = M^{-1} x\) transforms the system into two decoupled first-order DE: \[ \displaylines{\dot{z}_1 = \lambda_1 z_1 \\ \dot{z}_2 = \lambda_2 z_2 } \]

  1. \(\lambda_1 < 0, \lambda_2 < 0\): The equi. point \(x=0\) is stable. The phase portrait in \(z_1\)-\(z_2\) plane Stable Point

  2. \(\lambda_1 > 0, \lambda_2 > 0\): \(x=0\) is unstable. Change the arrow direction in the above image to get the phase portrait.

  3. \(\lambda_1 > 0 > \lambda_2\): \(x=0\) is a saddle point. Saddle Point

Case 2

\(\lambda_1 = \lambda_2 \in \mathbb{R}\), \(J = \begin{bmatrix} \lambda_1 & k \\ 0 & \lambda_2 \end{bmatrix}\), \(k\) is either 0 or 1

The phase portrait for \(k=0\) and \(k=1\) respectively:

Double Root Stable

Case 3

\(\lambda_{1,2} = \alpha \pm j \beta\), \(J = \begin{bmatrix} \alpha & -\beta \\ \beta & \alpha \end{bmatrix}\)

The phase portrait for \(\alpha < 0\), $> 0 $, \(\alpha = 0\) respectively:

Complex Root

\(x=0\) is referred as a stable focus if \(\alpha < 0\), unstable focus if \(\alpha > 0\), and center if \(\alpha = 0\)

Case 4

\(\lambda_1 \lambda_2 = 0\)

\(A\) has a nontrivial null space and the system has a equilibrium subspace.

Periodic Orbits

Consider the second-order autonomous system \[ \begin{equation}\displaylines{ \dot{x} = f(x) } \label{sys} \end{equation} \] where \(f(x)\) is cont. diff..

Poincare-Bendixson Criterion: Consider system \(\eqref{sys}\) and let \(M\) be a closed bounded subset of the plane s.t.

  • \(M\) contains no equi. points, or contains only one equi. point s.t. the Jacobian matrix \(\partial f / \partial x\) at this point has eigenvalues with positive real parts.
  • Every trajectory starting in \(M\) stays in \(M\) for all future time

Then, \(M\) contains a periodic orbit of \(\eqref{sys}\).

Bendixson Criterion: If, on a simply connected region \(D\) of the plane, \(\nabla \cdot f\) is not identically zero and does not change sign, then system \(\eqref{sys}\) has no periodic orbits lying entirely in \(D\)

Fundamental Properties

Definition

  • Connected set is a set that can not be partitioned into two open nonempty sets
  • Compact set: closed and bounded
  • Domain: open and connected set
  • Locally Lipschitz (LL) on a domain \(D \in \mathbb{R}^n\) if each point of \(D\) has a neighborhood \(D_0\) s.t. \(f\) satisfies the Lipschitz condition for all points in \(D_0\) with some Lipschitz const. \(L_0\).
  • Globally Lipschitz (GL): Lipschitz on \(\mathbb{R}^n\) with a uniform Lipschitz const..

Existence and Uniqueness

Theorem 1 (Local Existence and Uniqueness): Let \(f(t, x)\) be piecewise cont. in \(t\) and satisfy the Lipschitz condition \[ \displaylines{\| f(t, x) - f(t, y) \| \le L \| x - y \| } \] \(\forall x, y\in B = \{ x\in \mathbb{R}^n \mid \| x - x_0 \| \le r \}\), \(\forall t \in [t_0, t_1]\). Then there exists some \(\delta > 0\) s.t. the state equation \(\dot{x} = f(t, x)\) with \(x(t_0) = x_0\) has a unique solution over \([t_0, t_0 + \delta]\).

We have some lemmas below to prove Lipschitz condition by \(\partial f / \partial x\).

Lemma 1: Let \(f : [a, b] \times D \rightarrow \mathbb{R}^m\) be cont. on \(D\subseteq \mathbb{R}^n\). Suppose that \([\partial f / \partial x]\) exists and is cont. on \([a,b] \times D\). If, for a convex set \(W \subseteq D\), there is a const. \(L \ge 0\) s.t. \[ \displaylines{\left\| \frac { \partial f } { \partial x } ( t , x ) \right\| \leq L } \] on \([a,b] \times W\), then \[ \displaylines{\left\| f ( t , x ) - f ( t , y ) \right\| \leq L \| x - y \| } \] for all \(t \in [a,b]\), \(x\in W\), and \(y\in W\).

Lemma 2: If \(f(t,x)\) and \([\partial f / \partial x](t,x)\) are cont. on \([a,b]\times D\), for \(D\in \mathbb{R}^n\), then \(f\) is LL in \(x\) on \([a,b] \times D\).

Lemma 3: If \(f(t,x)\) and \([\partial f / \partial x](t,x)\) are cont. on \([a,b] \times \mathbb{R}^n\), then \(f\) is GL in \(x\) on \([a,b]\times \mathbb{R}^n\) iff \([\partial f / \partial x]\) is uniformly bounded (UB) on \([a,b]\times \mathbb{R}^n\).


Theorem 2 (Global Existence and Uniqueness): Let \(f(t,x)\) be piecewise cont. in \(t\) and satisfy \[ \displaylines{\| f(t, x) - f(t, y) \| \le L \| x - y \| } \] \(\forall x, y \in \mathbb{R}^n\), \(\forall t \in [t_0, t_1]\). Then, the state equation \(\dot{x} = f(t, x)\), with \(x(t_0) = x_0\), has a unique solution over \([t_0, t_1]\)


Theorem 3: Global existence and uniqueness theorem that requires \(f\) to be only LL:

Let \(f(t,x)\) be piecewise cont. in \(t\) and LL in \(x\) for all \(t \ge t_0\) and all \(x\) in \(D\subset \mathbb{R}^n\). Let \(W\) be a compact subset of \(D\), \(x_0 \in W\), and suppose every solution of \[ \begin{equation}\displaylines{ \dot{x} = f(t,x) \qquad x(t_0) = x_0 } \label{init} \end{equation} \] lies entirely in \(W\). Then, there is a unique solution that is defined for all \(t \ge t_0\).

Continuous Dependence on Initial Conditions and Parameters

The solution of \(\eqref{init}\) must depend cont. on the initial state \(x_0\), the initial time \(t_0\), and the right-hand side function \(f(t, x)\).


Theorem 4: Let \(f(t, x)\) be piecewise cont. in \(t\) and Lipschitz in \(x\) on \([t_0, t_1] \times W\) with a Lipschitz const. \(L\), where \(W \subset \mathbb{R}^n\) is an open connected set. Let \(y(t)\) and \(z(t)\) be the solution of \[ \displaylines{\dot{y} = f(t, y), \qquad y(t_0) = y_0 \\ \dot{z} = f(t, z) + g(t, z), \qquad z(t_0) = z_0 } \] s.t. \(y(t), z(t) \in W\) for all \(t \in [t_0, t)_1]\). Suppose that \[ \displaylines{\|g(t, x)\| \le \mu,\quad \forall (t, x) \in [t_0, t_1] \times W } \] for some \(\mu >0\), Then \[ \displaylines{\| y ( t ) - z ( t ) \| \leq \left\| y _{ 0 } - z_ { 0 } \right\| e^{L \left( t - t _{ 0 } \right)} + \frac { \mu } { L } \left( e^{ L \left( t - t_ { 0 } \right) } - 1 \right) } \] \(\forall t \in [t_0, t_1]\).


And the next theorem shows the continuity of solutions in terms of initial states and parameters.

Theorem 5: Let \(f(t, x, \lambda)\) be cont. in \((t, x, \lambda)\) and LL in \(x\) (uniformly in \(t\) and \(\lambda\)) on \([t_0, t_1] \times D \times \{ \|\lambda - \lambda_0 \| \le c \}\), where \(D \subset \mathbb{R}^n\) is an open connected set. Let \(y(t, \lambda_0)\) be a solution of \(\dot{x} = f(t, x, \lambda_0)\) with \(y(t_0, \lambda_0) = y_0 \in D\). Suppose \(y(t, \lambda_0)\) is defined and belongs to \(D\) for all \(t \in [t_0, t_1]\). Then, given \(\epsilon > 0\), there is \(\delta >0\) s.t. if \[ \displaylines{\| z_0 - y_0 \| < \delta, \qquad \| \lambda - \lambda_0 \| < \delta } \] then there is a unique solution \(z(t, \lambda)\) of \(\dot{x} = f(t, x, \lambda)\) defined on \([t_0, t_1]\), with \(z(t_0, \lambda) = z_0\), and \(z(t, \lambda)\) satisfies \[ \displaylines{\| z(t, \lambda) - y(t, \lambda_0) \| < \epsilon, \quad \forall t\in [t_0, t_1] } \]

Sensitivity Equations

Suppose that \(f(t, x, \lambda)\) is cont. in \((t, x, \lambda)\) and has cont. first partial derivatives w.r.t. \(x\) and \(\lambda\) for all \((t, x, \lambda) \in [t_0, t_1] \times \mathbb{R}^n \times \mathbb{R}^p\). Let \(\lambda_0\) be a nominal value of \(\lambda\), and suppose that the nominal state equation \[ \displaylines{\dot{x} = f(t,x,\lambda_0), \qquad x(t_0) = x_0 } \] has a unique solution \(x(t, \lambda_0)\) over \([t_0, t_1]\). We know that for all \(\lambda\) sufficiently close to \(\lambda_0\), the state equation \[ \displaylines{\dot{x} = f(t,x,\lambda), \qquad x(t_0) = x_0 } \] has a unique solution \(x(t, \lambda)\) over \([t_0, t_1]\) that is close to the nominal solution \(x(t, \lambda_0)\). The cont. diff. of \(f\) w.r.t. \(x\) and \(\lambda\) implies the additional property that the solution \(x(t, \lambda)\) is diff. w.r.t. \(\lambda\) near \(\lambda_0\). \[ \displaylines{x ( t , \lambda ) = x _{ 0 } + \int_ { t _{ 0 } } ^ { t } f ( s , x ( s , \lambda ) , \lambda ) d s } \] Take partial derivatives w.r.t. \(\lambda\) yields \[ \displaylines{x_ { \lambda } ( t , \lambda ) = \int _{ t_ { 0 } } ^ { t } \left[ \frac { \partial f } { \partial x } ( s , x ( s , \lambda ) , \lambda ) x _ { \lambda } ( s , \lambda ) + \frac { \partial f } { \partial \lambda } ( s , x ( s , \lambda ) , \lambda ) \right] d s } \] where \(x_\lambda (t_0, \lambda) = 0\). Differentiating w.r.t. \(t\) yields \[ \displaylines{\begin{aligned} \frac{\partial}{\partial t} x_\lambda (t, \lambda) &= \left.\frac{\partial f(t, x, \lambda)}{\partial x} \right|_{x = x(t, \lambda)} x_\lambda (t, \lambda) + \left.\frac{\partial f(t, x, \lambda)}{\partial \lambda} \right|_{x = x(t, \lambda)} \\ &= A(t, \lambda) x_\lambda (t, \lambda) + B(t, \lambda) \end{aligned} } \] For \(\lambda\) sufficiently close to \(\lambda_0\), the matrix \(A(t, \lambda)\) and \(B(t, \lambda)\) are defined on \([t_0, t_1]\). Hence, \(x_\lambda(t, \lambda)\) is defined on the same interval. Let \(S(t) = x_\lambda(t, \lambda_0)\), then \(S(t)\) is the unique solution of the equation \[ \begin{equation}\displaylines{ \dot{S}(t) = A(t, \lambda_0) S(t) + B(t, \lambda_0), \qquad S(t_0) = 0 } \label{sens} \end{equation} \] \(S(t)\) is called the sensitivity function, and \(\eqref{sens}\) is called the sensitivity equation.

Lyapunov Stability

Autonomous System

Consider the autonomous system \[ \begin{equation}\displaylines{ \dot{x} = f(x) } \label{as} \end{equation} \] where \(f:D\rightarrow \mathbb{R}^n\) is a LL map from \(D\subset \mathbb{R}^n\) into \(\mathbb{R}^n\). \(\bar{x}\) is an equilibrium point of the system if \(f(\bar{x}) = 0\). Without loss of generality, we can assume the equi. point is at the origin.


Definition 1: The equi. point \(x=0\) of \(\eqref{as}\) is

  • stable if, for each \(\epsilon >0\), there is \(\delta > 0\) s.t. \[ \displaylines{\| x(0) \| < \delta \Rightarrow \| x(t) \| < \epsilon, \quad\forall t \ge 0 } \]

  • unstable if it is not stable

  • asymptotically stable (AS) if it is stable and \(\delta\) can be chosen s.t. \[ \displaylines{\| x(0) \| < \delta \Rightarrow \lim_{t\to \infty} x(t) = 0 } \]


Theorem 1: Let \(x=0\) be an equi. point of \(\eqref{as}\) and \(D\subset \mathbb{R}^n\) be a domain containing \(x=0\). Let \(V : D \rightarrow \mathbb{R}\) be a cont. diff. function s.t. \[ \displaylines{\begin{aligned} V(0) &= 0 \\ V(x) &> 0 \quad \forall x \in D -\{0\} \\ \dot{V}(x) &\le 0 \quad \forall x \in D \end{aligned} } \] which is called a Lyapunov function. Then, \(x=0\) is stable. Moreover, if \[ \displaylines{\dot{V}(x) < 0 \quad \forall x \in D - \{0\} } \] then \(x=0\) is AS.


Theorem 2: Let \(x=0\) be an equi. point of \(\eqref{as}\). Let \(V : \mathbb{R}^n \rightarrow \mathbb{R}\) be a cont. diff. function s.t. \[ \displaylines{V(0) = 0 \\ V(x) > 0 \quad \forall x \ne 0 \\ \| x \| \rightarrow \infty \Rightarrow V(x) \rightarrow \infty \\ \dot{V}(x) < 0 \quad \forall x \ne 0 } \] then \(x=0\) is globally asymptotically stable (GAS).

A function \(V(x)\) satisfying the condition \(V(x) \rightarrow \infty\) as \(\| x \| \rightarrow \infty\) is said to be radially unbounded (RU).

If the origin is a GAS, then it must be the unique equi. point.


Theorem 3: Let \(x=0\) be an equi. point of \(\eqref{as}\). Let \(V: D \rightarrow \mathbb{R}\) be a cont. diff. function s.t. \(V(0) = 0\) and \(V(x_0) > 0\) for some \(x_0\) with arbitrarily small \(\| x_0 \|\). Define a set \(U=\{x \in B_r \mid V(x) > 0\}\) and suppose \(\dot{V}(x) > 0\) in \(U\). Then \(x=0\) is unstable.

Invariance Principle

Definition 2: A point \(p\) is said to be a positive limit point of \(x(t)\) if there is a sequence \(\{ t_n \}\), with \(t_n \to \infty\) as \(n\to \infty\), s.t. \(x(t_n) \to p\) as \(n \to \infty\).


Definition 3: A set \(M\) is said to be an invariant set w.r.t. \(\eqref{as}\) if \[ \displaylines{x(0) \in M \Rightarrow x(t) \in M, \quad \forall t \in \mathbb{R} } \] it is positively invariant set if \[ \displaylines{x(0) \in M \Rightarrow x(t) \in M, \quad \forall t \ge 0 } \]

Lemma 1: If a solution \(x(t)\) of \(\eqref{as}\) is bounded and belongs to \(D\) for \(t \ge 0\), then its positive limit set \(L^+\) is a nonempty, compact, invariant set. Moreover, \(x(t)\) approaches \(L^+\) as \(t \to \infty\).

Theorem 4: Let \(\Omega \subset D\) be a compact set that is PI w.r.t. \(\eqref{as}\). Let \(V : D \rightarrow \mathbb{R}\) be a cont. diff. function s.t. \(\dot{V}(x) \le 0\) in \(\Omega\). Let \(E=\{x \in \Omega \mid \dot{V}(x)=0 \}\). Let \(M\) be the largest invariant set in \(E\). Then every solution starting in \(\Omega\) approaches \(M\) as \(t \to \infty\).

Corollary 1: Let \(x=0\) be an equi. point of \(\eqref{as}\). Let \(V : D \rightarrow \mathbb{R}\) be a cont. diff. PD function on a domain \(D\) containing the origin, s.t. \(\dot{V}(x) \le 0\) in \(D\). Let \(S = \{ x \in D \mid \dot{V}(x) = 0 \}\) and suppose that no solution can stay identically in \(S\), other than the trivial solution \(x(t) \equiv 0\). Then the origin is AS.

Corollary 2: Let \(x=0\) be an equi. point of \(\eqref{as}\). Let \(V : \mathbb{R}^n \rightarrow \mathbb{R}\) be a cont. diff., RU, PD function s.t. \(\dot{V}(x) \le 0\) for all \(x \in \mathbb{R}^n\). Let \(S = \{ x \in \mathbb{R}^n \mid \dot{V}(x) = 0 \}\) and suppose that no solution can stay identically in \(S\), other than the trivial solution \(x(t) \equiv 0\). Then the origin is GAS.

LTI Systems and Linearization

Theorem 5: The equi. point \(x = 0\) of \(\dot{x} = Ax\) is stable iff all eigenvalues of \(A\) satisfy \(\operatorname{Re}\lambda_i \le 0\) and for every eigenvalues with \(\operatorname{Re}\lambda_i = 0\) and algebraic multiplicity \(q_i \ge 2\), \(\operatorname{rank}(A - \lambda_i I) = n - q_i\), where \(n\) is the dimension of \(x\). The equi. point \(x=0\) is (globally) AS iff all eigenvalues of \(A\) satisfy \(\operatorname{Re}\lambda_i < 0\)


Theorem 6: A matrix \(A\) is Hurwitz (\(\operatorname{Re}\lambda_i \le 0\)) iff for any given PSD \(Q\) there exists a PSD that satisfies the Lyapunov equation: \[ \displaylines{PA + A^T P = -Q } \] Moreover, if \(A\) is Hurwitz, then \(P\) is the unique solution.


Theorem 7 (Lyapunov's indirect method): Let \(x=0\) be an equi. point for \(\eqref{as}\), where \(f:D\rightarrow \mathbb{R}^n\) is cont. diff. and \(D\) is a neighborhood of the origin. Let \(A = \left. \frac{\partial f}{\partial x} \right|_{x=0}\), then

  • The origin is AS if \(\operatorname{Re}\lambda_i < 0\) for all eigenvalues of \(A\).
  • The origin is unstable if \(\operatorname{Re}\lambda_i > 0\) for at least one eigenvalue of \(A\).

Comparison Functions

Definition 4: A cont. function \(\alpha: [0, a) \rightarrow [0, \infty)\) is said to belong to \(\mathcal{K}\) if it is strictly increasing and \(\alpha(0) = 0\). It is said to belong to \(\mathcal{K}_\infty\) if \(a = \infty\) and \(\alpha(r) \rightarrow \infty\) as \(r \rightarrow \infty\).

Definition 5: A cont. function \(\beta: [0, a) \times [0, \infty) \rightarrow [0, \infty)\) is said to belong to \(\mathcal{KL}\) if, for each fixed \(s\), the mapping \(\beta(r, s)\) belongs to \(\mathcal{K}\) w.r.t. \(r\) and, for each fixed \(r\), the mapping \(\beta(r, s)\) is decreasing w.r.t. \(s\) and \(\beta(r, s) \rightarrow 0\) as \(s \rightarrow \infty\).


Lemma 2: Let \(\alpha_1\) and \(\alpha_2\) be \(\mathcal{K}\) functions on \([0, a)\), \(\alpha_3\) and \(\alpha_4\) be \(\mathcal{K}_\infty\) functions, and \(\beta\) be a \(\mathcal{KL}\) function. Denote the inverse of \(\alpha_i\) by \(\alpha_i^{-1}\). Then

  • \(\alpha_1^{-1}\) is defined on \([0, \alpha_1(a))\) and belongs to \(\mathcal{K}\)
  • \(\alpha_3^{-1}\) is defined on \([0, \infty)\) and belongs to \(\mathcal{K}_\infty\)
  • \(\alpha_1 \circ \alpha_2\) belongs to \(\mathcal{K}\)
  • \(\alpha_3 \circ \alpha_4\) belongs to \(\mathcal{K}_\infty\)
  • \(\sigma(r, s) = \alpha_1( \beta(\alpha_2(r), s))\) belongs to \(\mathcal{KL}\)


Lemma 3: Let \(V : D \rightarrow \mathbb{R}\) be a cont. PD function defined on a domain \(D \subset \mathbb{R}^n\) that contains the origin. Let \(B_r \subset D\) for some \(r > 0\). Then, there exist \(\mathcal{K}\) functions \(\alpha_1\) and \(\alpha_2\), defined on \([0, r]\), s.t. \[ \displaylines{\alpha_1(\|x\|) \le V(x) \le \alpha_2(\|x\|) } \] for all \(x\in B_r\). If \(D = \mathbb{R}^n\), the functions \(\alpha_1\) and \(\alpha_2\) will be defined on \([0, \infty)\) and the foregoing inequality will hold for all \(x\in \mathbb{R}^n\). Moreover, if \(V(x)\) is RU, then \(\alpha_1\) and \(\alpha_2\) can be chosen to belong to \(\mathcal{K}_\infty\)

Lemma 4: Consider the scalar autonomous DE \[ \displaylines{\dot{y} = -\alpha(y), \qquad y(t_0) = y_0 } \]

where \(\alpha\) is a LL \(\mathcal{K}\) function defined on \([0,a)\). For all \(0 \le y_0 \le a\), this equation has a unique solution \(y(t)\) defined for all \(t \ge t_0\). Moreover, \[ \displaylines{y(t) = \sigma(y_0, t_0) } \] where \(\sigma\) is a \(\mathcal{KL}\) function defined on \([0, a) \times [0, \infty)\).

Nonautonomous System

Consider the nonautonomous system \[ \begin{equation}\displaylines{ \dot{x} = f(t, x) } \label{nas} \end{equation} \] where \(f: [0, \infty) \times D \rightarrow \mathbb{R}^n\) is piecewise cont. in \(t\) and LL in \(x\) on \([0, \infty) \times D\), and \(D \subset \mathbb{R}^n\) is a domain that contains the origin. The origin is an equi. point at \(t=0\) if \[ \displaylines{f(t, 0) = 0, \quad \forall t \ge 0 } \]

Definition 6: The equi. point \(x=0\) of \(\eqref{nas}\) is

  • stable if, for each \(\epsilon >0\), there is \(\delta = \delta(\epsilon, t_0) > 0\) s.t. \[ \begin{equation}\displaylines{ \left\| x \left( t_ { 0 } \right) \right\| < \delta \Rightarrow \| x ( t ) \| < \varepsilon , \quad \forall t \geq t _ { 0 } \ge 0 } \label{nastable} \end{equation} \]

  • uniformly stable (US) if, for each \(\epsilon >0\), there is \(\delta = \delta(\epsilon) > 0\), independent of \(t_0\) s.t. \(\eqref{nastable}\) is satisfied.

  • unstable if it is not stable

  • AS if it is stable and there is a const. \(c=c(t_0) > 0\) s.t. \[ \begin{equation}\displaylines{ \| x(t_0) \| < c \Rightarrow \lim_{t\to \infty} x(t) = 0 } \label{naas} \end{equation} \]

  • uniformly asymptotically stable (UAS) if it is US and there exists \(c>0\), independent of \(t_0\) s.t. \(\eqref{naas}\) is satisfied.

  • globally uniformly asymptotically stable (GUAS) if it is US, \(\delta(\epsilon)\) can be chosen to satisfy \(\lim_{\epsilon \to \infty} \delta(\epsilon) = \infty\), and, for each pair of \(\eta > 0\) and \(c > 0\), there is \(T = T(\eta, c) > 0\) s.t. \[ \displaylines{\| x ( t ) \| < \eta , \quad \forall t \geq t_ { 0 } + T ( \eta , c ) , \quad \forall \left\| x \left( t _ { 0 } \right) \right\| < c } \]


Definition 7: The equi. point \(x=0\) of \(\eqref{nas}\) is

  • US iff there exist a \(\mathcal{K}\) function \(\alpha\) and a const. \(c>0\), independent of \(t_0\), s.t. \[ \displaylines{\| x ( t ) \| \leq \alpha \left( \left\| x \left( t_ { 0 } \right) \right\| \right) , \quad \forall t \geq t _{ 0 } \geq 0 ,\ \forall \left\| x \left( t_ { 0 } \right) \right\| < c } \]

  • UAS iff there exist a \(\mathcal{KL}\) function \(\beta\) and a const. \(c>0\), independent of \(t_0\), s.t. \[ \begin{equation}\displaylines{ \| x ( t ) \| \leq \beta \left( \left\| x \left( t_ { 0 } \right) \right\| , t - t _{ 0 } \right) , \quad \forall t \geq t_ { 0 } \geq 0 ,\ \forall \left\| x \left( t _ { 0 } \right) \right\| < c } \label{uaskl} \end{equation} \]

  • GUAS iff \(\eqref{uaskl}\) is satisfied for any initial state \(x(t_0)\)


Definition 8: The equi. point \(x=0\) of \(\eqref{nas}\) is exponentially stable (ES) if there exist const. \(c > 0\), \(k > 0\) and \(\lambda > 0\) s.t. \[ \displaylines{\| x ( t ) \| \leq k \left\| x \left( t _{ 0 } \right) \right\| e ^ { - \lambda \left( t - t_ { 0 } \right) } , \quad \forall \left\| x \left( t _{ 0 } \right) \right\| < c } \] and globally exponentially stable (GES) if it holds for any initial state \(x(t_0)\)


Theorem 8: Let \(x=0\) be an equi. point of \(\eqref{nas}\) and \(D\subset \mathbb{R}^n\) be a domain containing \(x=0\). Let \(V : [0, \infty) \times D \rightarrow \mathbb{R}\) be a cont. diff. function s.t. \[ \displaylines{W_1(x) \le V(t, x) \le W_2(x) \\ \frac{\partial V}{\partial t} + \frac{\partial V}{\partial x}f(t, x) \le 0 } \] \(\forall t \ge 0\) and \(\forall x \in D\), where \(W_1(x)\) and \(W_2(x)\) are cont. PD functions on \(D\). Then, \(x=0\) is US.


Theorem 9: Suppose the assumptions of Theorem 8 are satisfied with inequality strengthened to \[ \displaylines{\frac{\partial V}{\partial t} + \frac{\partial V}{\partial x}f(t, x) \le -W_3(x) } \] \(\forall t \ge 0\) and \(\forall x \in D\), where \(W_3(x)\) is a cont. PD function on \(D\). Then, \(x=0\) is UAS. Moreover, if \(r\) and \(c\) are chosen s.t. \(B_r = \{ \|x\| \le r \} \subset D\) and \(c < \min_{\|x\| = r} W_1(x)\), then every trajectory starting in \(\{ x \in B_r \mid W_2(x) \le c\}\) satisfies \[ \displaylines{\| x ( t ) \| \leq \beta \left( \left\| x \left( t_ { 0 } \right) \right\| , t - t _{ 0 } \right) , \quad \forall t \geq t_ { 0 } \geq 0 } \] for some class \(\mathcal{KL}\) function \(\beta\). Finally, if \(D = \mathbb{R}^n\) and \(W_1(x)\) is RU, then \(x=0\) is GUAS.


Theorem 10: Let \(x=0\) be an equi. point of \(\eqref{nas}\) and \(D\subset \mathbb{R}^n\) be a domain containing \(x=0\). Let \(V : [0, \infty) \times D \rightarrow \mathbb{R}\) be a cont. diff. function s.t. \[ \begin{equation}\displaylines{ { k _{ 1 } \| x \| ^ { a } \leq V ( t , x ) \leq k_ { 2 } \| x \| ^ { a } } \\ { \frac { \partial V } { \partial t } + \frac { \partial V } { \partial x } f ( t , x ) \leq - k _{ 3 } \| x \| ^ { a } } } \label{es} \end{equation} \] \(\forall t \ge 0\) and \(\forall x \in D\), where \(k_1, k_2, k_3, a\) are positive const.. Then \(x=0\) is ES. If the assumptions hold globally, then \(x=0\) is GES.

LTV Systems and Linearization

The stability for the LTV system \[ \begin{equation}\displaylines{ \dot{x}(t) = A(t) x } \label{ltv} \end{equation} \] can be completely characterized in terms of the state transition matrix \[ \displaylines{x(t) = \Phi(t, t_0) x(t_0) } \]

Theorem 11: The equi. point \(x=0\) of \(\eqref{ltv}\) is (globally) UAS iff the state transition matrix satisfies \[ \displaylines{\left\| \Phi \left( t , t _{ 0 } \right) \right\| \leq k e ^ { - \lambda \left( t - t_ { 0 } \right) } , \quad \forall t \geq t _ { 0 } \geq 0 } \] for some positive const. \(k\) and \(\lambda\)


Theorem 12: Let \(x=0\) be the ES equi. point of \(\eqref{ltv}\). Suppose \(A(t)\) is cont. and bounded. Let \(Q(t)\) be a cont., bounded, PD, symmetric matrix. Then, there is a cont. diff. bounded, PD, symmetric matrix \(P(t)\) that satisfies \[ \displaylines{-\dot { P } ( t ) = P ( t ) A ( t ) + A ^ { T } ( t ) P ( t ) + Q ( t ) } \] and \(V(t,x) = x^T P(t) x\) is a Lyapunov function for the system that satisfies \(\eqref{es}\)


Theorem 13: Let \(x=0\) be an equi. point for \(\eqref{nas}\), where \(f: [0, \infty) \times D \rightarrow \mathbb{R}^n\) is cont. diff., $D = { x |x|_2 < r } $, and the Jacobian matrix $ f / x $ is bounded and Lipschitz on \(D\), uniformly in $ t $. Let \[ \displaylines{A(t) = \left. \frac{\partial f}{\partial x} (t, x) \right|_{x=0} } \]

Then, the origin is ES for the nonlinear system iff it is an ES for the linear system \(\dot{x} = A(t) x\)

Converse Theorems

Theorem 14: Let \(x=0\) be an equi. point for \(\eqref{nas}\), where \(f: [0, \infty) \times D \rightarrow \mathbb{R}^n\) is cont. diff., $D = { x |x| < r } $, and the Jacobian matrix $ f / x $ is bounded on \(D\), uniformly in \(t\). Let \(k, \lambda, r_0\) be positive const. with \(r_0 < r/ k\). Let $D_0 = { x |x| < r_0 } $. Assume that the trajectories of the system satisfy \[ \displaylines{\| x ( t ) \| \leq k \left\| x \left( t _{ 0 } \right) \right\| e ^ { - \lambda \left( t - t_ { 0 } \right) } , \quad \forall x \left( t _{ 0 } \right) \in D_ { 0 } , \ \forall t \geq t _{ 0 } \geq 0 } \] Then, there is a function \(V: [0, \infty) \times D_0 \rightarrow \mathbb{R}\) that satisfies the inequalities \[ \displaylines{c _{ 1 } \| x \| ^ { 2 } \leq V ( t , x ) \leq c_ { 2 } \| x \| ^ { 2 } \\ \frac { \partial V } { \partial t } + \frac { \partial V } { \partial x } f ( t , x ) \leq - c _{ 3 } \| x \| ^ { 2 } \\ \left\| \frac { \partial V } { \partial x } \right\| \leq c_ { 4 } \| x \| } \] for some positive const. \(c_1, c_2, c_3, c_4\). Moreover, if \(r=\infty\) and the origin is GES, then \(V(t,x)\) is defined and satisfies the above inequalities on \(\mathbb{R}^n\). Furthermore, if the system is autonomous, \(V\) can be chosen independent of \(t\)


Theorem 15: Let \(x=0\) be an AS equil. point for \(\eqref{nas}\) where \(f: [0, \infty) \times D \rightarrow \mathbb{R}^n\) is cont. diff., $D = { x |x| < r } $, and the Jacobian matrix $ f / x $ is bounded on \(D\), uniformly in \(t\). Let \(\beta\) be a \(\mathcal{KL}\) function and \(r_0\) be a positive const. s.t. \(\beta(r_0, 0) < r\). Let \(D_0 = \{x\in \mathbb{R}^n \mid \|x\| < r_0 \}\). Assume that the trajectory of the system satisfies \[ \displaylines{\| x ( t ) \| \leq \beta \left( \left\| x \left( t _{ 0 } \right) \right\| , t - t_ { 0 } \right) , \quad \forall x \left( t _{ 0 } \right) \in D_ { 0 } ,\ \forall t \geq t _{ 0 } \geq 0 } \] Then, there is a cont. diff. function \(V: [0, \infty) \times D_0 \rightarrow \mathbb{R}\) that satisfies \[ \displaylines{\alpha _{ 1 } ( \| x \| ) \leq V ( t , x ) \leq \alpha_ { 2 } ( \| x \| ) \\ \frac { \partial V } { \partial t } + \frac { \partial V } { \partial x } f ( t , x ) \leq - \alpha _{ 3 } ( \| x \| ) \\ \left\| \frac { \partial V } { \partial x } \right\| \leq \alpha_ { 4 } ( \| x \| ) } \] where \(\alpha_{1,2,3,4}\) are \(\mathcal{K}\) functions defined on \([0, r_0]\). If the system is autonomous, \(V\) can be chosen independent of \(t\)


Theorem 16: Let \(x=0\) be an AS equil. point for \(\eqref{as}\) where \(f: D \rightarrow \mathbb{R}^n\) is LL and \(D \subset \mathbb{R^n}\) is a domain contains the origin. Let \(R_A \subset D\) be the region of attraction of \(x=0\). Then there is a smooth, PD function \(V(x)\) and a cont. PD function \(W(x)\), both defined for all \(x\in R_A\), s.t. \[ \displaylines{\lim_{x \rightarrow \partial R_ { A }} V ( x ) \rightarrow \infty \\ \frac { \partial V } { \partial x } f ( x ) \leq - W ( x ) , \quad \forall x \in R _{ A } } \] and for any \(c>0\), \(\{V(x) \le c\}\) is a compact subset of \(R_A\). When \(R_A = \mathbb{R}^n\), \(V(x)\) is RU.

Boundedness

Definition 9: The solution of \(\eqref{nas}\) are

  • uniformly bounded (UB) if there exists a positive const. \(c\), independent of \(t_0 \ge 0\), and for every \(a \in (0, c)\), there is \(\beta = \beta(a) > 0\), independent of \(t_0\), s.t. \(\left\| x \left( t _{ 0 } \right) \right\| \leq a \Rightarrow \| x ( t ) \| \leq \beta , \quad \forall t \geq t_ { 0 }\)
  • globally uniformly bounded (GUB) if UB holds for \(c=\infty\)
  • uniformly ultimately bounded (UUB) with ultimate bound \(b\) if there exist positive const. \(b\) and \(c\), independent of \(t_0 \ge 0\), and for every \(a\in (0, c)\), there is \(T = T(a, b) \ge 0\), independent of \(t_0\), s.t. \(\left\| x \left( t _{ 0 } \right) \right\| \leq a \Rightarrow \| x ( t ) \| \leq b , \quad \forall t \geq t_ { 0 } + T\)
  • globally uniformly ultimately bounded (GUUB) if UUB holds for \(c=\infty\)


Theorem 17: Let \(D \subset \mathbb{R}^n\) be a domain containing \(x=0\) and \(V : [0, \infty) \times D \rightarrow \mathbb{R}\) be a cont. diff. function s.t. \[ \displaylines{\alpha _{ 1 } ( \| x \| ) \leq V ( t , x ) \leq \alpha_ { 2 } ( \| x \| ) \\ \frac { \partial V } { \partial t } + \frac { \partial V } { \partial x } f ( t , x ) \leq - W _{ 3 } ( x ) , \quad \forall \|x\| \ge \mu > 0 } \] \(\forall t \ge 0\) and \(\forall x \in D\), where \(\alpha_{1,2}\) are \(\mathcal{K}\) functions and \(W_3(x)\) is a cont. PD function. Take \(r > 0\) s.t. \(B_r \subset D\) and suppose that \(\mu < \alpha_2^{-1}(\alpha_1(r))\), then, there exists a \(\mathcal{KL}\) function \(\beta\) and \(\forall x(t_0)\) that satisfies \(\|x(t_0)\| \le \alpha_2^{-1}(\alpha_1(r))\), there is \(T \ge 0\) (dependent on \(x(t_0)\) and \(\mu\)) s.t. the solution of \(\eqref{nas}\) satisfies \[ \displaylines{\| x ( t ) \| \leq \beta \left( \left\| x \left( t_ { 0 } \right) \right\| , t - t _{ 0 } \right) , \forall t_ { 0 } \leq t \leq t _{ 0 } + T \\ \| x ( t ) \| \leq \alpha_ { 1 } ^ { - 1 } \left( \alpha _{ 2 } ( \mu ) \right) , \forall t \geq t_ { 0 } + T } \] Moreover, if \(D = \mathbb{R}^n\) and \(\alpha_1\) belongs to \(\mathcal{K}_\infty\), then it holds \(\forall x(t_0)\)

Input-to-State Stability

Consider the system \[ \begin{equation}\displaylines{ \dot{x} = f(t, x, u) } \label{nasinput} \end{equation} \] where \(f: [0, \infty) \times \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n\) is piecewise cont. in \(t\) and LL in \(x\) and \(u\). The input \(u(t)\) is a piecewise cont., bounded function of \(t\) for all \(t \ge 0\). Suppose the unforced system \(\dot{x} = f(t, x, 0)\) has a GUAS equi. point at the origin. What can we say about the system \(\eqref{nasinput}\) in the presence of a bounded input \(u(t)\).

For the LTI system: \[ \displaylines{\dot{x} = Ax + Bu } \] with a Hurwitz matrix \(A\), we can write the solution as \[ \displaylines{x(t)=e^{\left(t-t_{0}\right) A} x\left(t_{0}\right)+\int_{t_{0}}^{t} e^{(t-\tau) A} B u(\tau) d \tau } \] and use the bound \(\left\|e^{\left(t-t_{0}\right) A}\right\| \leq k e^{-\lambda\left(t-t_{0}\right)}\) to estimate the solution by \[ \displaylines{\begin{aligned} \|x(t)\| & \leq k e^{-\lambda\left(t-t_{0}\right)}\left\|x\left(t_{0}\right)\right\|+\int_{t_{0}}^{t} k e^{-\lambda(t-\tau)}\|B\|\|u(\tau)\| d \tau \\ & \leq k e^{-\lambda\left(t-t_{0}\right)}\left\|x\left(t_{0}\right)\right\|+\frac{k\|B\|}{\lambda} \sup _{t_{0} \leq \tau \leq t}\|u(\tau)\| \end{aligned} } \] This estimate shows that the zero-input response decays to zero exponentially fast, while the zero-state response is bounded for every bounded input (BIBS).


Definition 10: The system \(\eqref{nasinput}\) is said to be input-to-state stable (ISS) if there exist a \(\mathcal{KL}\) function \(\beta\) and a \(\mathcal{K}\) function \(\gamma\) s.t. for any initial state \(x(t_0)\) and any bounded input \(u(t)\), the solution \(x(t)\) exists for all \(t \ge t_0\) and satisfies \[ \displaylines{\|x(t)\| \leq \beta\left(\left\|x\left(t_{0}\right)\right\|, t-t_{0}\right)+\gamma\left(\sup _{t_{0} \leq \tau \leq t}\|u(\tau)\|\right) } \]

Theorem 18: Let \(V : [0, \infty) \times \mathbb{R}^n \rightarrow \mathbb{R}\) be a cont. diff. function s.t. \[ \displaylines{\begin{aligned} \alpha_{1}(\|x\|) & \leq V(t, x) \leq \alpha_{2}(\|x\|) \\ \frac{\partial V}{\partial t}+\frac{\partial V}{\partial x} f(t, x, u) &\leq-W_{3}(x), \quad \forall\|x\| \geq \rho(\|u\|)>0 \end{aligned} } \] \(\forall (t,x,u) \in [0, \infty) \times \mathbb{R}^n \times \mathbb{R}^m\), where \(\alpha_{1,2}\) are \(\mathcal{K}_\infty\) functions, \(\rho\) is \(\mathcal{K}\) function, and \(W_3(x)\) is a cont. PD function on \(\mathbb{R}^n\). Then, the system \(\eqref{nasinput}\) is ISS with \(\gamma = \alpha_1^{-1} \circ \alpha_2 \circ \rho\)

Lemma 5: Suppose \(f(t,x,u)\) is cont. diff. and GL in \((x,u)\), uniformly in \(t\). If the unforced system \(\dot{x} = f(t, x, 0)\) has a GES equi. point at the origin, then the system \(\eqref{nasinput}\) is ISS.


Consider the cascade system: \[ \begin{align} dot{x}_{1}&=f_{1}\left(t, x_{1}, x_{2}\right) \label{cascade1} \\ \dot{x}_{2}&=f_{2}\left(t, x_{2}\right) \label{cascade2} \end{align} \] where \(f_1 : [0, \infty) \times \mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \rightarrow \mathbb{R}^{n_1}\) and \(f_2 : [0, \infty) \times \mathbb{R}^{n_2} \rightarrow \mathbb{R}^{n_2}\) are piecewise cont. in \(t\) and LL in \(x\). Suppose both \(\dot{x}_1 = f_1(t, x_1, 0)\) and \(\dot{x}_{2}=f_{2}\left(t, x_{2}\right)\) have GUAS equi. point at their respective origins.

Lemma 6: If the system \(\eqref{cascade1}\), with \(x_2\) as input, is ISS and the origin of \(\eqref{cascade2}\) is GUAS, then the origin of the cascade system is GUAS.

Input-Output Stability

\(\mathcal{L}\) Stability

Consider a system with input-output relation represented by \[ \displaylines{y = H u } \] Where \(H\) is a mapping from \(u\) to \(y\), \(u : [0, \infty) \rightarrow \mathbb{R}^m\).

Define the space \(\mathcal{L}_p^m\) for \(1\le p \le \infty\) as the set of all piecewise cont. functions \(u : [0, \infty) \rightarrow \mathbb{R}^m\) s.t. \[ \displaylines{\|u\|_{\mathcal{L}_{p}}=\left(\int_{0}^{\infty}\|u(t)\|^{p} d t\right)^{1 / p}<\infty } \]

Specifically, for \(p=2\) and \(p=\infty\), the space are defined respectively as \[ \displaylines{\begin{aligned} \|u\|_{\mathcal{L}_{2}} &= \sqrt{\int_{0}^{\infty} u^{T}(t) u(t) d t}<\infty \\ \| u \|_ { \mathcal { L } _{ \infty } } &= \sup_ { t \geq 0 } \| u ( t ) \| < \infty \end{aligned} } \] Define the extended space \(\mathcal{L}_e^m\) as \[ \displaylines{\mathcal{L}_{e}^{m}=\left\{u | u_{\tau} \in \mathcal{L}^{m}, \forall \tau \in[0, \infty)\right\} } \] where \(u_\tau\) is a truncation of \(u\) defined by \[ \displaylines{u_{\tau}(t)=\left\{\begin{array}{cc}{u(t),} & {0 \leq t \leq \tau} \\ {0,} & {t>\tau}\end{array}\right. } \]

Definition 1: A mapping \(H : \mathcal{L}_e^m \rightarrow \mathcal{L}_e^q\) is \(\mathcal{L}\) stable if there exist a \(\mathcal{K}\) function \(\alpha\), defined on \([0, \infty)\), and a nonnegative constant \(\beta\) s.t. \[ \displaylines{\left\|(H u)_{\tau}\right\|_{\mathcal{L}} \leq \alpha\left(\left\|u_{\tau}\right\|_{\mathcal{L}}\right)+\beta } \] for all \(u \in \mathcal{L}_e^m\) and \(\tau \in [0, \infty)\). It is finite-gain \(\mathcal{L}\) stable if there exist nonnegative const. \(\gamma\) and \(\beta\) s.t. \[ \displaylines{\left\|(H u)_{\tau}\right\|_{\mathcal{L}} \leq \gamma \left\|u_{\tau}\right\|_{\mathcal{L}} + \beta } \] for all \(u \in \mathcal{L}_e^m\) and \(\tau \in [0, \infty)\).

Note that the definition of \(\mathcal{L}_\infty\) stability is same as BIBO stability.


Definition 2: A mapping \(H : \mathcal{L}_e^m \rightarrow \mathcal{L}_e^q\) is small-signal \(\mathcal{L}\) stable (small-signal finite-gain \(\mathcal{L}\) stable) if there is a positive const. \(r\) s.t. inequality in definition 1 is satisfied for all \(u \in \mathcal{L}_e^m\) with \(\sup_{0 \le t \le \tau} \|u(t)\| \le r\)

\(\mathcal{L}\) Stability of State Models

Consider the system: \[ \begin{equation}\displaylines{ \begin{aligned} \dot{x} &=f(t, x, u), \quad x(0)=x_{0} \\ y &=h(t, x, u) \end{aligned} } \label{statemodel} \end{equation} \] where \(x\in \mathbb{R}^n\), \(u\in \mathbb{R}^m\), \(y\in \mathbb{R}^q\), \(f:[0, \infty) \times D \times D_u \rightarrow \mathbb{R}^n\) is piecewise cont. in \(t\) and LL in \((x,u)\); \(u:[0, \infty) \times D \times D_u \rightarrow \mathbb{R}^q\) is piecewise cont. in \(t\) and cont. in \((x,u)\); \(D \subset \mathbb{R}^n\) is a domain that contains \(x=0\), and \(D_u \subset \mathbb{R}^m\) is a domain that contains \(u = 0\). Suppose \(x=0\) is an equi. point of the unforced system \[ \begin{equation}\displaylines{ \dot{x} = f(t,x,0) } \label{unforced} \end{equation} \] Theorem 1: Consider the system \(\eqref{statemodel}\) and take \(r > 0\) and \(r_u > 0\) s.t. \(\{ \|x\| \le r \} \subset D\) and \(\{ \|u\| \le r_u \} \subset D_u\). Suppose that

  • \(x=0\) is an ES equi. ponit of \(\eqref{unforced}\), and there is a \(V(t, x)\) that satisfies \[ \displaylines{c_{1}\|x\|^{2} \leq V(t, x) \leq c_{2}\|x\|^{2} \\ \frac{\partial V}{\partial t}+\frac{\partial V}{\partial x} f(t, x, 0) \leq -c_{3}\|x\|^{2} \\ \left\|\frac{\partial V}{\partial x}\right\| \leq c_{4}\|x\| } \] for all \((t,x) \in [0, \infty) \times D\) for some positive const. \(c_{1,2,3,4}\)

  • \(f\) and \(h\) satisfy the inequalities \[ \begin{align} \|f(t, x, u)-f(t, x, 0)\| \leq L\|u\| \\ \|h(t, x, u)\| \leq \eta_{1}\|x\|+\eta_{2}\|u\| \label{hbound} \end{align} \] for all \((t,x,u) \in [0, \infty) \times D \times D_u\) for some nonnegative const. \(L, \eta_{1,2}\)

Then, for each \(x_0\) with \(\|x_0\| \le r \sqrt{c_1 / c_2}\), the system \(\eqref{statemodel}\) is small-signal finite-gain \(\mathcal{L}_p\) stable for each \(p \in [1, \infty]\). In particular, for each \(u \in \mathcal{L}_{pe}\) with \(\sup_{0 \le t \le \tau} \| u(t) \| \le \min \{ r_u, c_1 c_3 r / (c_2 c_4 L) \}\), the output \(y(t)\) satisfies \[ \begin{equation}\displaylines{ \left\|y_{\tau}\right\|_{\mathcal{L}_{p}} \leq \gamma \left\|u_{\tau}\right\|_{\mathcal{L}_{p}} + \beta } \label{outputgain} \end{equation} \] for all \(\tau \in [0, \infty)\), with \[ \displaylines{\gamma=\eta_{2}+\frac{\eta_{1} c_{2} c_{4} L}{c_{1} c_{3}}, \quad \beta=\eta_{1}\left\|x_{0}\right\| \sqrt{\frac{c_{2}}{c_{1}}} \rho, \text { where } \rho=\left\{\begin{array}{ll} {1,} & {\text { if } p=\infty} \\ {\left(\frac{2 c_{2}}{c_{3} p}\right)^{1 / p},} & {\text { if } p \in[1, \infty)} \end{array}\right. } \] Furthermore, if the origin is GES and all the assumptions hold globally (with \(D = \mathbb{R}^n\) and \(D_u = \mathbb{R}^m\)), then, for each \(x_0 \in \mathbb{R}^n\), the system \(\eqref{statemodel}\) if finite-gain \(\mathcal{L}_p\) stable for each \(p \in [1, \infty)\).

Corollary 1: Suppose that in some neighborhood of \((x=0, u=0)\), the function \(f(t,x,u)\) is cont. diff., the Jacobian matrices \(\partial f / \partial x\) and \(\partial f / \partial u\) are bounded, uniformly in \(t\), and \(h(t,x,u)\) satisfies \(\eqref{hbound}\). If the origin is an ES equi. point of \(\eqref{unforced}\), then there is a const. \(r_0 > 0\) s.t. for each \(x_0\) with \(\|x_0\| < r_0\), the system \(\eqref{statemodel}\) is small-signal finite-gain \(\mathcal{L}_p\) stable for each \(p \in [1, \infty]\). Furthermore, if all the assumptions hold globally and the origin is a GES equi. point of \(\eqref{unforced}\), then for each \(x_0 \in \mathbb{R}^n\), the system \(\eqref{statemodel}\) if finite-gain \(\mathcal{L}_p\) stable for each \(p \in [1, \infty]\)

Corollary 2: The LTI system \[ \begin{equation}\displaylines{ \begin{aligned} \dot{x} &=A x+B u \\ y &=C x+D u \end{aligned} } \label{ltiinput} \end{equation} \] is finite-gain \(\mathcal{L}_p\) stable for each \(p \in [1, \infty]\) if \(A\) is Hurwitz. Moreover, \(\eqref{outputgain}\) is satisfied with \[ \displaylines{\gamma=\|D\|_{2}+\frac{2 \lambda_{\max }^{2}(P)\|B\|_{2}\|C\|_{2}}{\lambda_{\min }(P)}, \quad \beta=\rho\|C\|_{2}\left\|x_{0}\right\| \sqrt{\frac{\lambda_{\max }(P)}{\lambda_{\min }(P)}}, \text { where } \rho=\left\{\begin{array}{ll} {1,} & {\text { if } p=\infty} \\ {\left(\frac{2 \lambda_\max(P)}{p}\right)^{1 / p},} & {\text { if } p \in[1, \infty)} \end{array}\right. } \] and \(P\) is the solution of the Lyapunov equation \(PA + A^TP = - I\)


Theorem 2: Consider the system \(\eqref{statemodel}\) and take \(r > 0\) s.t. \(\{ \|x\| \le r \} \subset D\). Suppose that

  • \(x=0\) is an UAS equi. ponit of \(\eqref{unforced}\), and there is a \(V(t, x)\) that satisfies \[ \displaylines{\alpha_{1}(\|x\|) \leq V(t, x) \leq \alpha_{2}(\|x\|) \\ \frac{\partial V}{\partial t}+\frac{\partial V}{\partial x} f(t, x, 0) \leq -\alpha_{3}(\|x\|) \\ \left\|\frac{\partial V}{\partial x}\right\| \leq \alpha_4(\|x\|) } \] for all \((t,x) \in [0, \infty) \times D\) for some \(\mathcal{K}\) functions \(\alpha_{1,2,3,4}\)

  • \(f\) and \(h\) satisfy the inequalities \[ \begin{align} \|f(t, x, u)-f(t, x, 0)\| \leq \alpha_5(\|u\|) \\ \|h(t, x, u)\| \leq \alpha_6(\|x\|) + \alpha_7(\|u\|) + \eta \label{hbound2} \end{align} \] for all \((t,x,u) \in [0, \infty) \times D \times D_u\) for some \(\mathcal{K}\) functions \(\alpha_{5,6,7}\), and a nonnegative const. \(\eta\)

Then, for each \(x_0\) with \(\|x_0\| \le \alpha_2^{-1}(\alpha_1(r))\), the system \(\eqref{statemodel}\) is small-signal \(\mathcal{L}_\infty\) stable.

Corollary 3: Suppose that in some neighborhood of \((x=0, u=0)\), the function \(f(t,x,u)\) is cont. diff., the Jacobian matrices \(\partial f / \partial x\) and \(\partial f / \partial u\) are bounded, uniformly in \(t\), and \(h(t,x,u)\) satisfies \(\eqref{hbound2}\). If the origin is an UAS equi. point of \(\eqref{unforced}\), then the system \(\eqref{statemodel}\) is small-signal \(\mathcal{L}_\infty\) stable.


Theorem 3: Consider the system \(\eqref{statemodel}\) with \(D = \mathbb{R}^n\) and \(D_u = \mathbb{R}^m\). Suppose that

  • The system \(\dot{x} =f(t, x, u), \quad x(0)=x_{0}\) is ISS
  • \(h\) satisfies \(\eqref{hbound2}\)

Then, for each \(x_0 \in \mathbb{R}^n\), the system \(\eqref{statemodel}\) is \(\mathcal{L}_\infty\) stable.

\(\mathcal{L}_2\) Gain

Theorem 4: Consider the system \(\eqref{ltiinput}\) where \(A\) is Hurwitz. Let \(G(s) = C (sI - A)^{-1} B + D\). Then, the \(\mathcal{L}_2\) gain of the system is \(\sup_{\omega \in \mathbb{R}} \| G(j \omega) \|_2\)


Theorem 5: Consider the time-invariant nonlinear system \[ \begin{equation}\displaylines{ \begin{aligned} \dot{x} &=f(x)+G(x) u, \quad x(0)=x_{0} \\ y &=h(x) \end{aligned} } \label{tinonlinear} \end{equation} \]

where \(f(x)\) is LL, and \(G(x), h(x)\) are cont. over \(\mathbb{R}^n\). The matrix \(G \in \mathbb{R}^{n \times m}\) and \(h : \mathbb{R}^n \rightarrow \mathbb{R}^q\). \(f(0)=0, h(0)=0\). Let \(\gamma\) be a positive number and suppose there is a cont. diff. PSD function \(V(x)\) that satisfies the Hamilton-Jacobi inequality \[ \displaylines{\mathcal{H}(V, f, G, h, \gamma) \stackrel{\text { def }}{=} \frac{\partial V}{\partial x} f(x)+\frac{1}{2 \gamma^{2}} \frac{\partial V}{\partial x} G(x) G^{T}(x)\left(\frac{\partial V}{\partial x}\right)^{T}+\frac{1}{2} h^{T}(x) h(x) \leq 0 } \] for all \(x \in \mathbb{R}^n\). Then, for each \(x_0 \in \mathbb{R}^n\), the system \(\eqref{tinonlinear}\) is finite-gain \(\mathcal{L}_2\) stable and its \(\mathcal{L}_2\) gain is less than or equal to \(\gamma\).

Corollary 4: Suppose the assumption of Theorem 5 are satisfied on a domain \(D \subset \mathbb{R}^n\) that contains the origin. Then, for any \(x_0 \in D\) and any \(u \in \mathcal{L}_{2e}\) for which the solution \(x\) of \(\eqref{tinonlinear}\) satisfies \(x(t) \in D\) for all \(t \in [0, \tau]\), we have \[ \displaylines{\left\|y_{\tau}\right\|_{\mathcal{L}_{2}} \leq \gamma\left\|u_{\tau}\right\|_{\mathcal{L}_{2}}+\sqrt{2 V\left(x_{0}\right)} } \]

Lemma 1: Suppose the assumption of Theorem 5 are satisfied on a domain \(D \subset \mathbb{R}^n\) that contains the origin, \(f(x)\) is a cont. diff function, and \(x=0\) is an AS equi. point of \(\dot{x} = f(x)\). Then, there is \(k_1 > 0\) s.t. for each \(x_0\) with \(\| x_0 \| \le k_1\), the system \(\eqref{tinonlinear}\) is small-signal finite-gain \(\mathcal{L}_2\) stable with \(\mathcal{L}_2\) gain less than or equal to \(\gamma\)

Lemma 2: Suppose the assumption of Theorem 5 are satisfied on a domain \(D \subset \mathbb{R}^n\) that contains the origin, \(f(x)\) is a cont. diff function, and no solution of \(\dot{x} = f(x)\) can stay identically in \(S = \{ x\in D | h(x) =0 \}\) other than \(x(t) \equiv 0\). Then, the origin of \(\dot{x} = f(x)\) is AS and there is \(k_1 > 0\) s.t. for each \(x_0\) with \(\| x_0 \| \le k_1\), the system \(\eqref{tinonlinear}\) is small-signal finite-gain \(\mathcal{L}_2\) stable with \(\mathcal{L}_2\) gain less than or equal to \(\gamma\)

Feedback Systems

Feedback System

Consider two systems \(H_1 : \mathcal{L}_e^m \rightarrow \mathcal{L}_e^q\) and \(H_2 : \mathcal{L}_e^q \rightarrow \mathcal{L}_e^m\). Suppose both systems are finite-gain \(\mathcal{L}\) stable, that is \[ \displaylines{{\left\|y_{1 \tau}\right\|_{\mathcal{L}} \leq} {\gamma_{1}\left\|e_{1 \tau}\right\|_{\mathcal{L}}+\beta_{1}, \quad \forall e_{1} \in \mathcal{L}_{e}^{m}, \forall \tau \in[0, \infty)} \\ {\left\|y_{2 \tau}\right\|_{\mathcal{L}}} {\leq \gamma_{2}\left\|e_{2 \tau}\right\|_{\mathcal{L}}+\beta_{2}, \quad \forall e_{2} \in \mathcal{L}_{e}^{q}, \forall \tau \in[0, \infty)} } \] Suppose further that the feedback system is well defined: for every pair of inputs \(u_1 \in \mathcal{L}_e^m\) and \(u_2 \in \mathcal{L}_e^q\), there exist unique outputs \(e_1, y_2 \in \mathcal{L}_e^m\) and \(e_2, y_1 \in \mathcal{L}_e^q\). Define \[ \displaylines{u=\begin{bmatrix}{u_{1}} \\ {u_{2}}\end{bmatrix}, \quad y=\begin{bmatrix}{y_{1}} \\ {y_{2}}\end{bmatrix}, \quad e=\begin{bmatrix} {e_{1}} \\ {e_{2}}\end{bmatrix} } \] The question is whether the feedback connection, when viewed as a mapping from \(u\) to \(e\) or a mapping from \(u\) to \(y\), is finite-gain \(\mathcal{L}\) stable. The two statements are equivalent.


Theorem 6: The feedback connection is finite-gain \(\mathcal{L}\) stable if \(\gamma_1 \gamma_2 < 1\).