BIBO STABILITY AND ASYMPTOTIC STABILITY

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1 BIBO STABILITY AND ASYMPTOTIC STABILITY FRANCESCO NORI Abstract. In this report with discuss the concepts of bounded-input boundedoutput stability (BIBO) and of Lyapunov stability. Examples are given to motivate the theoretical framework. 1. Introduction and motivation Many phenomena in nature can be modeled with the use of a dynamical systems. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems. In the following we give two examples; the first introduces a discrete time dynamical system; the second a continuous time one. Example 1.1 (Discrete time population model). Let us consider a dynamical model which describes the number of individuals within a population. Let s divide the population into two classes, nominally elder people and younger people. Let x 1 (t) be the number of elder individuals at time t = 0, 1, 2,... ; let x 2 (t) be the number of young individuals and u(t) be the immigrants which are assumed to belong only to the younger class. Obviously, the total number of individuals in the population is given by y(t) = x 1 (t) + x ( 2). A possible dynamical system which describes the population growing across time is the following: (1.1) (1.2) where: and where: } x 1 (t + 1) = βx 2 (t) : state transition x 2 (t + 1) = α 1 x 1 (t) + α 2 x 2 (t) + u(t) y(t) = x 1 (t) + x 2 (t) } x 1 (0) = x 1,0 initial condition. x 2 (0) = x 2,0 β: survival factor α 1 : nativity factor for the elder individuals α 2 : nativity factor for the younger individuals x1 : x 2 system state u(t): system input y(t): system output The model is such that given the initial condition x 1,0, x 2,0 and the evolution of the input u(t), t = 0, 1, 2..., it is possible to determine x 1 (t) and x 2 (t) for every time sample t = 0, 1, 2... Interesting questions can be posed given the model. Suppose for example that the number of individuals is limited, i.e. 0 < u(t) < u max for every 1

2 2 FRANCESCO NORI time sample t. Can we then guarantee that the population y(t) = x 1 (t) + x 2 (t) will also be limited across time, i.e. 0 < y(t) < y max for every time sample t? What it (surprisingly) turns out is that the answer can be no for certain choice of the parameters β, α 1 and α 2. Example 1.2 (Continuous time predator-prey model: Lodka-Volterra). Let us consider a dynamical model which describes the number of predators and preys within an isolated ecosystem. The number of preys at time t is denoted with the variable x 1 (t) while the number of predators is denoted with the variable x 2 (t). A possible description of the system dynamical evolution is given by the following dynamical system: (1.3) (1.4) where: and: 1 dt (t) = αx 1(t) βx 1 (t)x 2 (t) 2 dt (t) = γx 2(t) + δx 1 (t)x 2 (t) y(t) = x 1 (t) + x 2 (t) } x 1 (0) = x 1,0 initial condition. x 2 (0) = x 2,0 } : state transition α: prey nativity rate γ: predator mortality rate β: prey mortality rate due to predators δ: predator nativity x1 : x 2 system state y(t): system output The above dynamical system has an equilibrium in x 1 = γ δ = x 1,eq, x 2 = α β = x 2,eq. Practically, if for some t we have x 1 (t ) = x 1,eq and x 2 (t ) = x 2,eq, then: { 1 dt = αx 1,eq βx 2,eq x 2,eq = 0 2 dt = γx 2,eq + δx 1,eq x 2,eq = 0, and therefore the number of predators and preys will remain constantly x 1,eq, y 2,eq, i.e.: (1.6) x 1 (t) = x 1,eq, x 2 (t) = x 2,eq t > t. Given the interesting properties of the equilibrium point, it results natural to ask whether the equilibrium point is attractive, i.e. if the number of preys and the number of predators will always end up to be x 1,eq and x 2,eq, respectively. Sophisticate analysis tools that will be presented in the following show that the equilibrium is indeed attractive. Example 1.3 (Continuous time car model). Let s consider the following dynamical model: m d2 p (1.7) (t) = bdp(t) + u(t), dt2 dt p(0) = p 0, dp dt (0) = ṗ 0 which can be used to model the forward motion of a car in presence of a viscous friction (see 1 pages 24 to 27). Specifically we have:,

3 STABILITY 3 m: mass of the car p(t): car position p(0): car initial dp dt (t): car velocity dt (t2 ): car acceleration b: viscous friction coefficient u(t): force produced by the engine d 2 p The dynamic model above can be rearranged as follows: 1 dt (t) = x } 2(t) 2 dt (t) = bx : state transition (1.8) 2(t) + u(t) y(t) = x 1 (t) (1.9) where: (1.10) and: } x 1 (0) = p 0 initial condition. x 2 (0) = ṗ 0 u(t): y(t): x 1 (t) = p(t) x 2 (t) = dp dt (t). y(t) = p(t) x1 : system state x 2 system input system output The above dynamical equations are such that given u(t), t 0, + there exists a unique output y(t), t 0, + ); the proof of this claim is non-trivial and the interested reader should refer to BIBO stability: continuous time case In the previous example we have seen that a continuous dynamical system can be seen as a map from the input function u(t), t 0, + ) to the output function y(t), t 0, + ). In this section we propose necessary and sufficient conditions for guaranteeing that a given continuous dynamical system is BIBO stable. Roughly speaking, BIBO stability assures that a limited input will lead to a limited output Linear, causal, time-invariant SISO systems. Let us a consider a generic map A from the input function u(t), t 0, + ) 1 to the output function y(t), t (, + ). These maps are the so called single-input single-output (SISO) systems. Practically: (2.1) u : 0, + ) R To simplify the notation we will write y = A(u). A y : (, + ) R., 1 We here assume that the function u(t) is identically null for t < 0, that is u(t) = 0, t (, 0.

4 4 FRANCESCO NORI Definition 2.1 (Past of a function with respect to the time instant T ). Given a function f(t), t, + ) and a scalar T R its past P T f with respect to the time instant T is a function (P T f)(t), t, T ) defined as follows: { f(t) if t < T (2.2) (P T f)(t) =. 0 if t T Definition 2.2 (Causality of a SISO map). We say that a map A of type (2.1) is causal if and only if P T (Au) = P T A(P T u) for all possible values of T. Roughly speaking a system is causal if the value of the output at time t is determined only by the values assumed by the input in the time instants before t. Equivalently, y(t) is determined only by u(τ) with τ 0, t. Remark 2.3. It can be shown that if the map (2.1) is causal then y = Au is such that y(t) = 0 for any t < 0. This is a consequence that u(t) = 0 for any t < 0. Definition 2.4 ( -delayed function). Given a function f(t), t (, + ) and a scalar R, the -delayed function σ f is a function (σ f)(t), t (, + ) defined as follows: (2.3) (σ f)(t) = f(t + ) t (, + ). Definition 2.5 (Time invariance of a SISO map). We say that a map A of type (2.1) is time invariant if and only if A(σ u) = σ (Au) for all possible values of T. Roughly speaking a system is time invariant if applying the same input at different time instants produces the same response. Definition 2.6 (Linear SISO map). We say that a map A of type (2.1) is linear if and only if: A(α 1 u 1 + α 2 u 2 ) = α 1 (Au 1 ) + α 2 (Au 2 ), for all α 1, α 2, u 1 and u 2. Roughly speaking a system is linear if scaling the input produces a scaled output. At the same time, summing two inputs produces an output which is the sum of the two outputs produced by applying the inputs individually. Remark 2.7. If a SISO map is linear, then Au = 0 if u(t) for all t 0, + ). The proof is trivial and therefore left to the reader. In the previous definitions we characterized SISO systems in terms of linearity, causality and time-invariance. A map which can be proven to linear, causal and time-invariant is convolution integral. Let h(t) be a function defined for positive t i.e. h : 0, + R. The convolution integral associated to h is given by: (2.4) y(t) = (Au)(t) = t It can be shown that the map described by: (2.5) u : 0, + ) R 0 h(t σ)u(σ)dσ = (h u)(t). A y(t) = t 0 h(t σ)u(σ)dσ is linear, causal and time-invariant. What it turns out is that the converse is also true. Loosely speaking, any SISO map of type (2.1) which is linear, causal, timeinvariant map can be described by a convolution integral (2.5) for a suitably chosen h.

5 STABILITY 5 Remark 2.8. In order to express any linear, causal, time-invariant system as a convolution integral it is necessary expand the set of the considered h. Consider for example the identity map y = Au = u which returns an output which is identical to the input. Obviously this map is linear, causal and time-invariant. However, it cannot be expressed as a convolution integral (2.5) if we only restrict h to be a function. A broader class must be considered, nominally the set of distributions D R. The interested reader should refer to a more advance textbook. Theorem 2.9. If a map A : D R D R is linear, causal and time-invariant, then it can be expressed as a convolution integral Au = h u with h D R BIBO stability of SISO systems. We here give necessary and sufficient conditions for guaranteeing that a given linear, causal, time-invariant map of type (2.1) is BIBO stable. Definition 2.10 (BIBO stability). We say that a given map of type (2.1) is BIBO stable if and only if for every limited input the output is itself limited. Mathematically: (2.6) sup t u(t) < + sup y(t) < + with y = Au. t Theorem Consider a linear, causal, time-invariant map A : D R D R. Let h D R be a description of the same map with a convolution integral, i.e.: (2.7) Au = h u u D R. The following statements are equivalent: (2.8) The map A is BIBO stable. The function h(t) is such that: + 0 h(t) dt < BIBO stability of SISO systems with rational transfer function. Consider a linear, causal, time-invariant map A : D R D R whose convolution integral is described by h D R. Let H(s), s C be the Laplace transform of the convolution function h. The function H(s) is usually called the transfer function associated to A. We here consider a special class of maps, nominally those that are described by rational transfer functions, i.e.: (2.9) H(s) = n(s) d(s) = α 0 + α 1 s + + α m s m β 0 + β 1 s + + β n s n. Practically, we consider systems whose transfer functions are the fraction of two polynomial. For this specific class of systems the following theorem holds. Theorem Consider a linear, causal, time-invariant map A : D R D R and suppose that its transfer function, H(s), is rational i.e.: (2.10) H(s) = n(s) d(s) = α 0 + α 1 s + + α m s m β 0 + β 1 s + + β n s n. The following statements are equivalent: The map A is BIBO stable.

6 6 FRANCESCO NORI (2.11) The roots λ 1,..., λ n of the polynomial d(s) are all in the left half complex plane, i.e.: Re(λ i ) < 0, i = 1, 2,... n, where Re(z) denotes the real part of the complex number z. Theorem Consider a BIBO stable dynamical system described by the convolution y = h u with transfer function H(s). Let the input be: (2.12) u(t) = A sin(ωt + φ) Then the output y(t) is such that: (2.13) t 0, + ). y(t) t + A W (iω) sin(ωt + φ + argw (iω)). where W (jω) and argw (iω)) are respectively the modulus and the argument of the complex number W (iω). Practically, the steady state output is again a sinusoid scaled by W (jω) and delayed by argw (iω)). Remark The above theorem can be used to approximate the steady state output when the input u(t) is periodic. Using the Fourier approximation we have: (2.14) u(t) N A n sin(nωt + φ n ) n=0 and using the linearity of the system we have: N (2.15) y(t) t + A n W (inω) sin(nωt + φ n + argw (inω)). n=0 Example Consider a linear, causal, time-invariant map whose transfer function is: (2.16) s + 1 H(s) = (s + 1)(s + 2). Since the transfer function is rational we can use the previous theorem and check the BIBO stability by computing the roots λ 1, λ 2 of the denominator polynomial d(s) = (s + 1)(s + 2). We have λ 1 = 1, λ 2 = 2 and therefore the system is BIBO stable. Example Consider a linear, causal, time-invariant map whose transfer function is: (2.17) s + 1 H(s) = s 2 + 2s + 3. Again, the BIBO stability can be checked by computing the roots λ 1, λ 2 of the denominator polynomial d(s) = s 2 + 2s + 3. We have: (2.18) λ 1,2 = b ± b 2 4ac 2a = 2 ± = 1 ± 2i. Therefore, we have λ 1 = 1+ 2i and λ 2 = 1 2i. Since, Re(λ 1 ) = Re(λ 2 ) = 1 the system is BIBO stable.

7 STABILITY 7 3. Continuous time dynamical system Definition 3.1. A continuous time dynamical system is a differential relation of the form: (3.1) where: y(t) = and: { dt x 1 (t). x n (t) y 1 (t). y p (t) = f(x(t), u(t)) y(t) = h(x(t), u(t)) x(0) = x 0, R n : state vector u(t) = R p : output vector x 0 = f : R n R m R n : h : R n R p R n : u 1 (t). u m (t) x 1,0. x n,0 t 0, + ) R m : R n : state transition function output function input vector initial state vector We will here consider a restrict class of dynamical systems, nominally linear dynamical systems. Linear systems are those characterized by a linear state transition function and a linear output function, i.e.: (3.2) { dt = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) x(0) = x 0. The system (3.2) is a special case of linear, causal, time-invariant system when we make the following two additional assumptions: SISO (single input single output) systems, i.e. systems such that the input u(t) and the output y(t) are scalars. the system initial condition is zero 2, i.e. x 0 = 0, so that u(t) = 0, t 0, + ) implies y(t) = 0, t 0, + ). Under these two assumptions, the system can be expressed as a convolution integral. The transfer function of the system turns out to be (see 3): (3.3) H(s) = C(sI A) 1 B + D which can be easily proven to be rational. Example 3.2. Let us consider the dynamical model (1.8) described in Example 1.3. The system can be easily written in the form (3.2) with the following definitions: x1 (t) 0 1 0, A =, B =, C = x 2 (t) 0 b 1 1 0, D = 0. 2 This assumption is necessary to guarantee that y = 0 if u(t) = 0 for all t in 0, + ).

8 8 FRANCESCO NORI Let s compute the system transfer function: H(s) = C(sI A) 1 B = s s + b 1 = s s(s+b) 0 1 = 1 s(s + b). 0 1 s+b The denominator polynomial is d(s) = s(s+b) which has roots λ 1 = 0 and λ 2 = b. Since Re(λ 1 ) = 0 the system is not BIBO stable. However, if we consider a new input ũ(t) = u(t) + kx 1 (t) we obtain: (3.4) with: { dt = Ax(t) + Bũ(t) y(t) = Cx(t) x1 (t) 0 1 0, A =, B =, C = x 2 (t) k b 1 1 0, D = 0. Easy computations lead to: H(s) = 1 s 2 + bs + k. Taking k = 3 and b = 2 the system is BIBO stable with respect to the new input ũ (compare with Example 2.16). More in general, it can be proven that the system is BIBO stable for any k > 0 and b > Lyapunov stability: continuous time case As opposed to BIBO stability, we here consider the concept of Lyapunov stability which is not related to the input-output properties of a dynamical system. Specifically, Lyapunov stability applies only to unforced (no control input) dynamical systems. It is used to study the behavior of dynamical systems under initial perturbations around equilibrium points. Consider an unforced dynamical system described by the following differential equation 3 : (4.1) dt (t) = f(x(t)) x(0) = x 0. Definition 4.1 (Equilibrium point). We say that x eq is an equilibrium point for the system (4.1) if f(x eq ) = 0. Remark 4.2. If at time t the system state x falls into an equilibrium point, then it will remain in the equilibrium point for t > t. In particular if x 0 = x eq, then x eq, t 0, + )., 3 Such a system can be obtained from (3.1) considering an identically null input, i.e. u(t) = 0, for every t 0, + ).

9 STABILITY 9 The Lyapunov theoretical framework helps in characterizing a given equilibrium point. In particular, it helps in discriminating two different situations, nominally stable equilibria and unstable equilibria. Roughly speaking, an equilibrium point is said to be stable if when the system is displaced from the equilibrium there are forces that drive the system back to the equilibrium (see Figure 1). Otherwise the equilibrium is said to be unstable (see Figure 3). Figure 1. The picture shows the intuitive idea of stable equilibrium point. Figure 2. The picture shows the intuitive idea of unstable equilibrium point. Definition 4.3 (Stable equilibrium point). Consider the system (3.1) and let x eq be an equilibrium point. The equilibrium point x eq is said to be stable if for every neighborhood U of x eq there exists a neighborhood V such that for every initial condition x 0 V the system trajectory remains in U, i.e. x(t) U, for all t 0, + ). Definition 4.4 (Unstable equilibrium point). Consider the system (3.1) and let x eq be an equilibrium point. The equilibrium point x eq is said to be unstable if it is not stable. Definition 4.5 (Asymptotically stable equilibrium point). Consider the system (3.1) and let x eq be an equilibrium point. The equilibrium point x eq is said to be asymptotically stable if it is stable and for every initial condition the trajectory tends to x eq, i.e. x(t) t + x eq. Example 4.6 (Second order harmonic oscillator). We here consider a very simple oscillator, made of a mass m attached to a spring of stiffness k. Let p be the

10 10 FRANCESCO NORI Figure 3. The picture shows the intuitive idea of unstable equilibrium point. Figure 4. Spring. position of the mass in a fixed reference frame. written as follows (4.2) d 2 p dt 2 (t) = ω2 p(t) p(0) = p 0, dp dt (0) = ṗ 0. The system dynamics can be where ω 2 = k/m. In this specific case the trajectory of the system can be explicitly computed, and therefore we can easily determine if a given equilibrium is stable. Let s first write the dynamics in the form (4.1). We have: (4.3) dt (t) = x2 (t) ω 2 x 1 (t) x(0) = p0 ṗ 0, where we have chosen: p(t) (4.4) dp dt (t). Given the initial condition the trajectory followed by the system is: p0 cos(ωt) + ṗ0 ω sin(ωt) (4.5). ωp 0 sin(ωt) + ṗ 0 cos(ωt) Notice that 0 0 = xeq is an equilibrium point and we can therefore ask if it is stable. All we have to prove is that the trajectories of the system remain in an arbitrarily small neighborhood of x eq. Indeed, it can be easily seen that this can be achieved choosing a small enough initial condition p 0, ṗ 0. Therefore, the equilibrium is stable. Example 4.7. We here consider a simple system whose dynamics are described as follows: (4.6) d 2 p dt 2 (t) = ω2 p(t) p(0) = p 0, dp dt (0) = ṗ 0.

11 STABILITY 11 Once again, in this specific case the trajectory of the system can be explicitly computed, and therefore we can easily determine if a given equilibrium is stable. Let s first write the dynamics in the form (4.1). We have: dt (t) = x2 (t) p0 (4.7) ω 2 x(0) =, x 1 (t) ṗ 0 where we have chosen: p(t) (4.8) dp dt (t). Given the initial condition the trajectory followed by the system is: ( ) ( ) 1 2 p 0 + ṗ0 ω e ωt p 0 ṗ0 ω e (4.9) ωt 1 2 (ωp 0 + ṗ 0 ) e ωt 1 2 (ωp. 0 ṗ 0 ) e ωt Notice that 0 0 = xeq is an equilibrium point and we can therefore ask if it is stable. However, the system trajectory always diverges for suitably chosen initial conditions. Therefore the equilibrium is unstable. Example 4.8 (Damped second order harmonic oscillator). We here consider again the oscillator in Figure 4 but adding a viscous friction b. The system dynamics can be written as follows d 2 p (4.10) dt 2 (t) = ω2 p(t) β dp dt (t) p(0) = p 0, dp dt (0) = ṗ 0. where ω 2 = k/m and β = b/m. Let s first write the dynamics in the form (4.1). We have: (4.11) dt (t) = x 2 (t) ω 2 x 1 (t) βx 2 (t) x(0) = p0 ṗ 0. To simplify the computations let s assume β = 2ω. Given the initial condition the trajectory followed by the system is: p 0 e ωt + (ṗ 0 + ωp 0 )te (4.12) ωt ωp 0 e ωt + (ṗ 0 + ωp 0 )e ωt ω(ṗ 0 + ωp 0 )te ωt. Notice that 0 0 = xeq is an equilibrium point and we can therefore ask if it is stable. It can be easily seen that this can be achieved choosing a small enough initial condition p 0, ṗ 0. Moreover, the equilibrium is asymptotically stable since x(t) t + x eq Lyapunov stability theorem. In this section we given a fundamental tool for characterizing an equilibrium without explicitly computing the trajectories of the considered system. Theorem 4.9 (Lyapunov stability theorem). Let x eq be an equilibrium point for the differential equation described by: (4.13) dt (t) = f(x(t)) x(0) = x 0. If there exists a function V : R n R such that: V has a minimum in x eq, i.e. V (x) > V (x eq ) for all x R n ;

12 12 FRANCESCO NORI The function V (x) defined as: (4.14) V (x) = x f(x), x = x 1... x n is negative semi-definite, i.e.: (4.15) V (x) 0 x R n. Then, the equilibrium x eq is stable and the function V (x) is said to be a Lyapunov function for the given system. Proof. Let s first observe that evaluated on a system trajectory x(t), the function V (x) equals the time derivative of the function V (x(t)), i.e.: (4.16) 0 V (x(t)) = d dt V (x(t)). We now prove that given a neighborhood U, we can find a neighborhood W such that if x 0 W then x(t) U for every t. Given U let V min be the minimum value assumed by V on the border of U. Now choose a neighborhood W so that the maximum value assumed by V on W is less then V min. We claim that if x 0 W then x(t) U for every t. Suppose by contradiction that the system crosses the border of U at time t > 0. Therefore V (x( t)) V min. At the same time since x 0 W, we have V (x(0)) < V min. Using the two conditions we have V (x( t)) > V (x(0)) with t > 0 which is incompatible with d dtv (x(t)) 0. Theorem 4.10 (Lyapunov asymptotic stability theorem). Let x eq be an equilibrium point for the differential equation described by: (4.17) dt (t) = f(x(t)) x(0) = x 0. If there exists a function V : R n R such that: V has a minimum in x eq, i.e. V (x) > V (x eq ) for all x R n ; The function V (x) defined as: (4.18) V (x) = x f(x), x = x 1... x n is negative definite, i.e.: (4.19) V (x) < 0 x R n, x x eq Proof. See 4. Then the equilibrium is asymptotically stable. Example 4.11 (Second order harmonic oscillator). We here consider a the oscillator, proposed in Example 4.6. Let the candidate Lyapunov function be: (4.20) V (x) = 1 2 x ω2 x 2 1, which clearly satisfies V (x) > 0, for all x x eq. We have: (4.21) and since: (4.22) x = x 1 x 2 = x 1 f(x) = x2 x 2 = ω 2 x 1 x 2, ω 2 x 1

13 STABILITY 13 we get: (4.23) V (x) = ω 2 x x 1 x 2 2 ω 2 = 0, x 1 which is identically zero. Therefore, the given equilibrium is stable. We cannot say that the equilibrium is asymptotically stable because V (x) is not strictly less then zero. Example We here consider a damped oscillator like (see Example 4.8). Let the candidate Lyapunov function be: V (x) = 1 2 x ω2 x (4.24) (x 2 + 2ωx 1 ) 2 + ω 2 x which clearly satisfies V (x) > 0, for all x x eq. We have: x = (4.25) and since: (4.26) we get: x 1 x 2 = x 1 x 2 = 6ω 2 x 1 + 2ωx 2 2ωx 1 + 2x 2, x f(x) = 2 ω 2 x 1 2ωx 2 V (x) = 6ω 2 x 1 + 2ωx 2 2ωx 1 + 2x 2 x 2 ω 2 x 1 2ωx 2 = 2ω 3 x 2 1 2ωx 2 2, which clearly satisfies V (x) < 0 for every x x eq. Therefore, the given equilibrium is asymptotically stable. References 1. G. Franklin, J. Powell, and A. Emami-Naeini, Feedback control of dynamic systems, Addison- Wesley Publishing Company, Alberto Isidori, Nonlinear control systems: Third edition, Springer, T. Kailath, Linear systems, Prentice Hall, J. E. Slotine and W. Li, Applied nonlinear control, Prentice-Hall International Eds., address: iron@liralab.it