Introduction to Control Theory
LEMMA 2.5–2: Given the linear system H such that an input u(t) results in
2.6 Stability Theory
The first stability concept we study, concerns the behavior of free systems, or equivalently, that of forced systems with a given input. In other words, we study the stability of an equilibrium point with respect to changes in the initial conditions of the system. Before doing so however, we review some basic definitions. These definitions will be stated in terms of continuous, nonlinear systems with the understanding that discrete, nonlinear systems admit similar results and linear systems are but a special case of nonlinear systems.
Let xe be an equilibrium (or fixed) state of the free continuous-time, possibly time-varying nonlinear system
(2.6.1) i.e. f(xe, t)=0, where x, f are n×1 vectors.
We will first review the stability of an equilibrium point xe with the understanding that the stability of the state x(t) can always be obtained with a translation of variables as discussed later. The stability definitions we use can be found in [Khalil 2001], [Vidyasagar 1992].
DEFINITION 2.6–1 In all parts of this definition xe is an equilibrium point at time t0, and ||.|| denote any function norm previously defined.
1. Stability: xe is stable in the sense of Lyapunov (SL) at t0, if starting
xe is stable in the sense of Lyapunov if it is stable for any given t0. See Figure 2.6.1a.
2. Instability: xe is unstable in the sense of Lyapunov (UL), if no matter how close to xe the state starts, it will not be confined to the vicinity of xe at some later time. In other words, xe is unstable if it is not stable at t0. See Figure 2.6.1b for an illustration.
3. Convergence: xe is convergent (C) at t0, if states starting close to xe
will eventually converge to xe. In other words, xe is convergent at t0
if for any positive there exists a positive 1(t0) and a positive T(1, x0, t0) such that if
then
Figure 2.6.1: (a) Stability of xe at t0; (b) Instability of xe at t0.
53 xe is convergent, if it is convergent for any t0. See Figure 2.6.2 for illustration.
Figure 2.6.2: Convergence of xe at t0
4. Asymptotic Stability: xe is asymptotically stable (AS) at t0 if states starting sufficiently close to xe will stay close and will eventually converge to it. More precisely, xe is AS at t0 if it is both convergent and stable at t0. xe is AS if it is AS for any t0. An illustration of an AS equilibrium point is shown in Figure 2.6.3.
Figure 2.6.3: Asymptotic stability of xe at t0
5. Global Asymptotic Stability: xe is globally asymptotically stable (GAS) at t0 if any initial state will stay close to xe and will eventually converge to it. In other words, xe is GAS if it is stable at t0, and if every x(t) converges to xe as time goes to infinity. xe is GAS if it is 2.6 Stability Theory
EXAMPLE 2.6–1: Stability of Various Systems
1. Consider the scalar time-varying system given by
the solution of this equation for all tt0 is
The equilibrium point is located at xe=ye=0. Let us use the 1-norm given by |y| and suppose that our aim is to keep |y(t)|< for all tt0. It can be seen that our objective is achieved if
The origin is therefore a stable equilibrium point of this system.
This is further illustrated in Figure 2.6.5
2. The damped pendulum system has many equilibrium points as described in Example 2.3.1. It can be shown that the equilibrium point located at the origin of the state-space is unstable. This is illustrated in Figure 2.6.6 where it is seen that no matter how close to the origin the initial state is, the norm of x(t) can not be pre-specified. On the other hand, note that the two equilibrium points at [, 0] are stable.
Figure 2.6.4: Global asymptotic stability of xe at t0
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3. The origin is an equilibrium point of the Van der Pol oscillator.
However, and as shown in Figure 2.6.7, it is an unstable equilibrium point. In fact, suppose the following norm is used per definition 2.5.6, and let ≠1.Therefore, we would like
for all t>t0. As can be seen from Figure 2.6.7, no matter how close to the origin x0 is, i.e. no matter how small is, the trajectory will eventually leave the ball of radius =1.
4. The origin is a stable equilibrium point of the robot described in Example 2.3.1 whenever the following choices are made
Kv=diag(kvi); Kp=diag(kpi) where kvi>0 and kpi>0 for all i=1,…n.
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EXAMPLE 2.6–2: Stability versus Convergence Consider the following system
Figure 2.6.5: Time history for 2.6 Stability Theory
Figure 2.6.6: Damped pendulum: (a)time history; (b) phase plane
There are 2 equilibrium points located at (0, 0) and (1, 0), both of which are unstable (check!). On the other hand, (1, 0) is convergent since all trajectories will eventually converge to it after some time T. Before T however, there is no guarantee that a trajectory will stay within some of (1, 0) no matter how close the initial state is to (1, 0). In fact, suppose the system starts at x(0), then the state-vector is given at any t0 by
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See Figure 2.6.8 for illustration of the behavior of the state vector.
䊏 Note that stability and asymptotic stability are local concepts in the sense that, if the initial perturbation is too large, the subsequent states x(t) may stray arbitrarily far from xe. There exists, therefore, a region
Figure 2.6.7: Van der Pol oscillator: (a) time history; (b) phase plane 2.6 Stability Theory
centered at xe and given by such that both stability and asymptotic stability will result for any state starting in but not for states starting outside of it. This region is called the domain of attraction of xe. The equilibrium state xe is GAS if . Note also that all previous stability definitions depended on the initial time t0, so that the region of attraction may vary with varying initial times. If the system (2.6.1) were independent of time (or autonomous), then the stability concepts in definition 2.6.1 are indeed independent of t0 and they
Figure 2.6.8: Example 2.6.2 (a)time history; (b) phase plane
59 will be equivalent to the stability concepts defined next. On the other hand, and even though the system (2.6.1) is time-dependent, we would like to have its stability properties not depend on t0 since that would later provide us with a desired degree of robustness. This leads us to define the uniform stability concepts [Khalil 2001].
DEFINITION 2.6–2 In all parts of this definition, xe is an equilibrium point at time t0.
1. Uniform Stability: xe is uniformly stable (US) over [t0, ∞) if δ(, t0)in definition 2.6.1 is independent of t0.
2. Uniform Convergence: xe is uniformly convergent (UC) over [t0, ∞) if δ1(t0) and T(1,x0, t0) of definition 2.6.1 can be chosen independent of t0.
3. Uniform Asymptotic Stability: xe is uniformly, asymptotically stable (UAS) over [t0, ∞), if it is both US and UC.
4. Global Uniform Asymptotic Stability: xe is globally, uniformly, asymptotically stable (GUAS) if it is US, and UC.
5. Global Exponential Stability: xe is globally exponentially stable (GES) if there exists α>0, and ß 0 such that for all x0∈ℜn,
䊏 Note that GES implies GUAS, and see Figure 2.6.9 for an illustration of uniform stability concepts.
EXAMPLE 2.6–3: Uniform stability 1. Consider the damped Mathieu equation,
The origin is a US equilibrium point as shown in Figure 2.6.10 2. The scalar system
has an equilibrium point at the origin which is UC.
2.6 Stability Theory
3. The origin is a UAS equilibrium point for
4. The system
has an equilibrium point xe=0 which is GUAS.
5. Consider the system
the origin is then a GES equilibrium point since the solution is given by Figure 2.6.9: (a) Uniform Stability of xe; (b) uniform convergence of xe; (c) uniform asymptotic stability of xe; (d) global uniform asymptotic stability of xe
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so that
|x(t)|≤|x0|e-t
See Figure 2.6.11, for an illustration of the time history of x(t).
Figure 2.6.10: Damped Mathieu equation: (a)time history; (b) phase plane 䊏 2.6 Stability Theory
In many cases, a bound on the size of the state is all that is required in terms of stability. This is a less stringent requirement than Lyapunov stability. It is instructive to study the subtle difference between the definition of Boundedness below and that of Lyapunov stability in Definition 2.6.1.
DEFINITION 2.6–3
1. Boundedness: xe is bounded (B) at t0 if states starting close to xe will never get too far. In other words, xe is bounded at t0 if for each δ>0 such that
||x0-xe||<δ
there exists a positive (r, t0)<∞ such that for all tt0
xe is bounded if it is bounded for any t0.
2. Uniform Boundedness: xe is uniformly bounded (UB) over [t0, ∞) if (r, t0) can be made independent of t0.
3. Uniform Ultimate Boundedness: xe is said to be uniformly, ultimately bounded (UUB), if states starting close to xe will eventually become
Figure 2.6.11: Example 2.6.3e: (a)time history; (b) phase plane
63 bounded. More precisely, xeis UUB if for any δ>0, >0, there exists a finite time T(,δ) such that whenever ||x0-xe||<δ, the following is satisfied
for all tT(,δ).
4. Global Uniform Ultimate Boundedness: xeis said to be globally, uniformly, ultimately bounded (GUUB) if for >0, there exists a finite time T() such that
for all tT() See Figure 2.6.12 for an illustration of the boundedness stability concepts.
EXAMPLE 2.6–4: Boundedness 1. The second-order system given by
has a uniformly bounded equilibrium point at the origin as shown in Figure 2.6.13.
2. The second-order system given by
has an UUB equilibrium point at xe=0, as shown in Figure 2.6.14.
Note that in general, we are interested in the stability of the motion x(t) when the system is perturbed from its trajectory. In other words, how far does x(t) get from its nominal trajectory if the initial state is perturbed?
This problem can always be reduced to the stability of the origin of a non-䊏
䊏 2.6 Stability Theory
autonomous system by letting
z=xe-x(t) and
and studying the stability of the equilibrium point ze=0.
Figure 2.6.12: (a) Boundedness of xe at t0; (b) uniform boundedness of xe; (c) uniform ultimate boundedness of xe; (d) global uniform boundedness of xe.
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EXAMPLE 2.6–5: Stability of the Origin
1. Consider the damped pendulum of Example 2.3.1a. Its equilibrium points are at [nπ 0]T, n=0, ±1,…. The stability of these points can be studied from the stability of the origin of the system
2. Consider the rigid robot equations of Example 2.3.2, and assume Figure 2.6.13: Example 2.6.4-a: (a)x1(0)=x2(0)=1 (b) x1(0)=x2(0)=0.1
2.6 Stability Theory
that a desired trajectory is specified by
Therefore, we can define the new system by choosing z=xd-x so that Figure 2.6.14: Example 2.6.4-b: (a)x1(0)=x2(0)=1 (b) x1(0)=x2(0)=1
67 and verify that ze=0 is the desired equilibrium point of the modified system if xe=xd is the desired equilibrium trajectory of the robot.
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