Introduction to Control Theory64 autonomous system by letting z=x e -x(t) and and studying the stability of the equilibrium point z e =0. Figure 2.6.12: (a) Boundedness of x e at t 0 ; (b) uniform boundedness of x e ; (c) uniform ultimate boundedness of x e ; (d) global uniform boundedness of x e . Copyright © 2004 by Marcel Dekker, Inc. 65 EXAMPLE 2.6–5: Stability of the Origin 1. Consider the damped pendulum of Example 2.3.1a. Its equilibrium points are at [n π 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)x 1 (0)=x 2 (0)=1 (b) x 1 (0)=x 2 (0)=0.1 2.6 Stability Theory Copyright © 2004 by Marcel Dekker, Inc. Introduction to Control Theory66 that a desired trajectory is specified by Therefore, we can define the new system by choosing z=x d -x so that Figure 2.6.14: Example 2.6.4-b: (a)x 1 (0)=x 2 (0)=1 (b) x 1 (0)=x 2 (0)=1 Copyright © 2004 by Marcel Dekker, Inc. 67 and verify that z e =0 is the desired equilibrium point of the modified system if x e =x d is the desired equilibrium trajectory of the robot. 2.7 Lyapunov Stability Theorems Lyapunov stability theory deals with the behavior of unforced nonlinear systems described by the differential equations (2.7.1) where without loss of generality, the origin is an equilibrium point of (2.7.1). It may seem to the reader that such a theory is not needed since all we had to do in the examples of the previous section is to solve the differential equations, and study the time evolution of a norm of the state vector. There are at least two reasons why Lyapunov theory is needed. The first is that Lyapunov theory will allow us to determine the stability of a particular equilibrium point without actually solving the differential equations. This, as is well known to any student of nonlinear differential equations, is a large saving. The second and related reason for using Lyapunov theory is that it provides us with qualitative results to the stability questions, which may be used in designing stabilizing controllers of nonlinear dynamical systems. We shall first assume that any necessary conditions for (2.7.1) to have a unique solution are satisfied [Khalil 2001], [Vidyasagar 1992]. The unique solution corresponding to x(t 0 )=x 0 is x(t, t 0 , x 0 ) and will be denoted simply as x(t). Before we actually introduce Lyapunov’s theorems, we review certain classes of functions which will simplify the statement of Lyapunov theorems. Functions of Class K Consider a continuous function α:ℜ→ℜ DEFINITION 2.7–1 We say that a belongs to class K, if 1. α (0)=0 2. α (x)>0, for all x>0 3. α is nondecreasing, i.e. α (x 1 )Ն α (x 2 ) for all x 1 >x 2 . 2.7 Lyapunov Stability Theorems Copyright © 2004 by Marcel Dekker, Inc. Introduction to Control Theory68 EXAMPLE 2.7–1: Class K Functions The function α (x)=x 2 is a class K function. The function α (x)=x 2 +1 is not a class K function because (1) fails. On the other hand, α (x)=-x 2 is not a class K function because (2) and (3) fail. DEFINITION 2.7–2 In the following, ℜ + =[0, ∞). 1. Locally Positive Definite: A continuous function V:ℜ + ×ℜ n →R is locally positive definite (l.p.d) if there exists a class K function a(.) and a neighborhood N of the origin of ℜ n such that V(t, x)Ն α (||x||) for all tՆ0, and all x∈N. 2. Positive Definite: The function V is said to be positive definite (p.d) if N=ℜ n . 3. Negative and Local Negative Definite: We say that V is (locally) negative definite (n.d) if -V is (locally) positive definite. EXAMPLE 2.7–2: Locally Positive Definite Functions [Vidyasagar 1992] The function is l.p.d but not p.d, since V(t, x)=0 at x=(0, π/2). On the other hand, is not even l.p.d because V(t, x)→0 as t→∞ for any x. The function is p.d. DEFINITION 2.7–3 In the following, 1. Locally Decrescent: A continuous function is locally decrescent if There exists a class K function ß(.) and a neighborhood N of the origin of such that V(t, x)Յß(||x||) for all tՆ0 and all x∈ Ν. Copyright © 2004 by Marcel Dekker, Inc. 69 2. Decrescent: We say that V is decrescent if N=ℜ n . EXAMPLE 2.7–3: Decrescent Functions [Vidyasagar 1992] The function is locally but not globally decrescent. On the other hand, is globally decrescent. DEFINITION 2.7–4 Given a continuously differentiate function V: ℜ + ×ℜ n → R together with a system of differential equations (2.7.1), the derivative of V along (2.7.1) is defined as a function V: ℜ + ×ℜ n →R given by EXAMPLE 2.7–4: Lyapunov Functions Consider the function of Example 2.7.3 and assume given a system Then, the derivative of V(t, x) along this system is Lyapunov Theorems We are now ready to state Lyapunov Theorems, which we group in Theorem 2.7.1. For the proof, see [Khalil 2001], [Vidyasagar 1992]. THEOREM 2.7–1: Lyapunov Given the nonlinear system 2.7 Lyapunov Stability Theorems Copyright © 2004 by Marcel Dekker, Inc. Introduction to Control Theory70 with an equilibrium point at the origin, i.e. f(t, 0)=0, and let N be a neighborhood of the origin of size ⑀ i. e. Then 1. Stability: The origin is stable in the sense of Lyapunov, if for x∈ Ν , there exists a scalar function V(t, x) with continuous partial derivative such that (a) V(t, x) is positive definite (b) V is negative semi-definite 2. Uniform Stability: The origin is uniformly stable if in addition to (a) and (b) V(t, x) is decrescent for x∈ Ν. 3. Asymptotic Stability: The origin is asymptotically stable if V(t, x) satisfies (a) and is negative definite for x∈ Ν. 4. Global Asymptotic Stability: The origin is globally, asymptotically stable if V(t, x) verifies (a), and V(t, x) is negative definite for all x∈ℜ n i.e. if N=ℜ n . 5. Uniform Asymptotic Stability: The origin is UAS if V(t, x) satisfies (a), V(t, x) is decrescent, and V(t,x) is negative definite for x∈ Ν. 6. Global Uniform Asymptotic Stability: The origin is GUAS if N=ℜ n , and if V(t, x) satisfies (a), V(t,x) is decrescent, V(t,x) is negative definite and V(t, x) is radially unbounded, i.e. if it goes to infinity uniformly in time as ||x||→∞. 7. Exponential Stability: The origin is exponentially stable if there exists positive constants α , ß, γ such that 8. Global Exponential Stability: The origin is globally exponential stable if it is exponentially stable for all x∈ℜ n . The function V(t, x) in the theorem is called a Lyapunov function. Note that the theorem provides sufficient conditions for the stability of the origin and that the inability to provide a Lyapunov function candidate has no indication on the stability of the origin for a particular system. Copyright © 2004 by Marcel Dekker, Inc. 71 EXAMPLE 2.7–5: Stability via Lyapunov Functions 1. Consider the system described by and choose a Lyapunov function candidate Then the origin may be shown to be a stable equilibrium point. 2. Consider the Mathieu equation described in Example 2.6.3. Let the Lyapunov function candidate be given by The origin is then shown to be a US equilibrium point. 3. The system given in Example 2.6.3–5, has a GES equilibrium point at the origin. This may be shown by considering a Lyapunov function candidate V(x)=x 2 which leads to Then, 0.5x 2 ՅV(x)Յ2x 2 and The above inequalities hold for any x∈ℜ n . 2.7 Lyapunov Stability Theorems Copyright © 2004 by Marcel Dekker, Inc. Introduction to Control Theory72 4. Let and pick a Lyapunov function candidate so that so that the origin is SL. Lyapunov Theorems may be used to design controllers that will stabilize a nonlinear system such as a robot. In fact, if one chooses a Lyapunov function candidate V(t, x), then finding its total derivative V(t, x) will exhibit an explicit dependence on the control signal. By choosing the control signal to make V(t, x) negative definite, stability of the closed-loop system is guaranteed. Unfortunately, it is not always easy to guarantee the global asymptotic stability of an equilibrium point using Lyapunov Theorem. This is due to the fact, that V(t, x) may be shown to be negative but not necessarily negative-definite. If the open-loop system were autonomous, Lyapunov theory is greatly simplified as shown in the next section. The Autonomous Case Suppose the open-loop system is not autonomous, i.e. is not explicitly dependent on t, then a time-independent Lyapunov function candidate V(x) may be obtained and the positive definite conditions are greatly simplified as described next. LEMMA 2.7–1: A time invariant continuous function V(x) is positive definite if V(0)=0 and V(x)>0 for x≠0. It is locally positive definite if the above holds in a neighborhood of the origin Note that the condition that V(0)=0 is not necessary and that as long as V(0) is bounded above the Lyapunov results hold without modification. Copyright © 2004 by Marcel Dekker, Inc. [...]... if kvi and kpi are both positive Assume for the purposes of illustration that kvi =3 and kpi=2, and that ui=sin (t) Note that ui is bounded and let us find the output yi(t) yi(t) =-0 .2e-2t+0.5e-t-0 .32 cos(t+0 .32 ) which is bounded above by 0.62 and below by -0 .02 The derivative of y(t) is also bounded On the other hand, suppose the input is ui(t)=e-3t then the output is yi(t)=0.5e-t-e-2t+0.5e-3t Since... present the KY and MKY lemmas The MKY Lemma The following lemmas are versions of the Meyer-Kalman-Yakubovich (MKY) lemma which appears in [Narendra and Taylor 19 73] , [Khalil 2001] amongst other places, and will be useful in designing adaptive controllers for robots LEMMA 2.9–1: Meyer-Kalman-Yakubovitch Let the system (2.9 .3) with D=0 be controllable Then the transfer function c(sI-A )-1 b is SPR if and only... both observable and controllable These compensators are known as the observer-controller compensators and are shown for example in Figure 2.11.1 In the SISO case, a transfer function admits a state-space realization which is completely observable and controllable if and only if no pole-zero cancellations occur [Kailath 1980], [Antsaklis and Michel 1997] The next example shows an observer-controller compensator... Taylor 19 73] In fact, let us describe positive-real systems and discuss some of their properties Consider the multi-input-multi-output linear time-invariant system where x is an n vector, u is an m vector, y is a p vector, A, B, C, and D are of the appropriate dimensions The corresponding transfer function matrix is P(s)=C(sI-A )-1 B+D We will assume that the system has an equal number of inputs and outputs,... Dekker, Inc 90 Introduction to Control Theory 2 f(t, 0)=0 i.e the origin is an equilibrium point of f(t, x) 3 || f(t, x1)-f(t, x2)||Յß1||x1-x2||, for some ß1>0 4 ||g(t, x1)||Յß2r, for some ß2>0 5 ||g(t, x1)-g(t, x2)Յß2 ||x1-x2|| 6 Then, there exists a unique solution x(t) to 1.5.5 and The total stability theorem will be used to design controllers that will make the linear part of the system exponentially... uniformly bounded 2.11 Linear Controller Design The purpose of control design is to make the robot respond in a predictable and desirable fashion to a set of input signals In this section we review how a linear controller may be designed in order that the behavior of the robotcontroller combination is acceptable It seems obvious that the firs requirement on the robot- controller is its stability Therefore,... u=-Kx+v will place the eigenvalues of A-bk anywhere in the s plane The state-feedback gain needed to place the eigenvalues may be found in the single-input case using Ackermann’s formula (see for example [Antsaklis and Michel 1997]) Assume the desired closed-loop eigenvalues are specified as the roots of the equation (2.11 .3) Then, the state-feedback controller is given by (2.11.4) where C is the controllability... rankC=rank , and controllability is therefore preserved under state-space transformation By combining the concepts of observability and controllability, we can design compensators that solve the Linear Control Design problem In fact, the following theorem summarizes linear control design THEOREM 2.11–4: The Linear Control Design problem is solvable for a system 2.11.1, if and only if the state-space realization... lemma which relaxes the condition of controllability is given next LEMMA 2.9–2: Meyer-Kalman-Yakubovitch Given vector b, an asymptotically stable A, a real vector v, scalars γՆ0 and ⑀>0, and a positivedefinite matrix Q, then, there exist a vector q and a symmetric positive definite P such that 1 2 if and only if 1 is small enough and, 2 the transfer function γ/2+vT(sI-A )-1 b is SPR In most of our applications,... eigenvalues of A have negative real parts or that A has one eigenvalue at the origin while the rest of them is in the open left-half plane (OLHP), and 2 (A, b) is controllable, and 3 (c, A) is observable, and 4 The nonlinearity φ(.,.) satisfies (a) φ(t,0)=0;᭙tՆ0 and (b) Then, find conditions on the linear system (A, B, C, D) such that x=0 is a GAS equilibrium point of the closed-loop system Note that sometimes . is strictly less than zero for all Sometimes, and although V(x) is only non-positive, LaSalle’s theorem [LaSale and Lefschetz 1961], [Khalil 2001] may be used to guarantee the global asymptotic. such that 8. Global Exponential Stability: The origin is globally exponential stable if it is exponentially stable for all x∈ℜ n . The function V(t, x) in the theorem is called a Lyapunov function Lyapunov function candidate is so that and the origin is SL and actually USL. EXAMPLE 2.7–7: Uniform Stability via Lyapunov Functions This example illustrates the local asymptotic, uniform stability of