EML5311 Lyapunov Stability & Robust Control Design

Transcription

1 EML5311 Lyapunov Stability & Robust Control Design 1 Lyapunov Stability criterion In Robust control design of nonlinear uncertain systems, stability theory plays an important role in engineering systems. For any given control system, it is crucial to have a stable system since an unstable control system is useless. Lyapunov 1. stability criterion is a general and useful procedure for studying the stability of nonlinear systems. The Lyapunov stability theory include two methods, Lyapunov s first method and Lyapunov s direct method. Lyapunov s first method is a technique which simply uses the idea of system linearization(lowest order approximation) around a given point and one can only achieve local stability results with small stability regions. Lyapunov s direct method is the most important tool for design and analysis of nonlinear systems. Lyapunov s direct method is directly applied to nonlinear systems without the need to linearization and thus achieves global stability. The basic concept behind Lyapunov s direct method is that if the total energy of a system, electrical/mechanical; linear/nonlinear, is continuously dissipating, then the system will eventually reach an equilibrium point and remain at that point. Hence, Lyapunov s direct method include two steps, first find a appropriate scalar function, referred to as Lyapunov function, second evaluate its first-order time derivative along the trajectory of the system. If the Lyapunov function derivative is decreasing along the system trajectory as time increases, then the system energy is dissipating and thus the system will eventually settle down. The definitions below give a more formal statement of admissible choices of Lyapunov function candidate. Autonomous systems: the nonlinear system ẋ = f(x, u, t) is said to be autonomous if f does not depend explicitly on time, i.e., if the system can be written ẋ = f(x) Otherwise, the system is called non-autonomous. Equilibrium point: A state x e is an equilibrium point(state) of the system if x(t) = x e, 1 theory introduced in late 19 th century by the Russian mathematician Alexandr Mikhailovich Lyapunov 1

2 then it remains equal to x e for all time. Mathematically, this means that x e satisfies 0 = f(x e ). In this paper, we are mainly interested in stability of equilibrium points. Stability and instability: The equilibrium point x e = 0 is said to be stable if, for any Γ > 0, there exists γ > 0, such that if x(0) < γ, then x(t) < Γ for all t 0. Otherwise, the equilibrium point is unstable. Asymptotic stability: An equilibrium point 0 is asymptotically stable if it is stable, and if in addition there exits some γ > 0 such that x(0) < γ implies that x(t) 0 as t. Exponential stability: An equilibrium point 0 is exponentially stable if there exits two strictly positive numbers α and β such that x(t) α x(0) e βt, t > 0, in some ball B γ in the neighborhood of the origin. Lyapunov s first method: 1. The equilibrium point of the nonlinear system is asymptotically stable if the linearized system is strictly stable. 2. The equilibrium point of the nonlinear system is unstable if the linearized system is strictly unstable. 3. If the linearized system is marginally stable, one cannot conclude anything from the linear approximation (equilibrium point may be stable, unstable, or asymptotically stable for the nonlinear system) Lyapunov function: If function V(x) is positive definite and has continuous partial derivatives in a ball B γ, and if its time derivative along any state trajectory of system ẋ = f(x) is negative semi-definite, i.e., V (x 0, then V (x) is said to be a Lyapunov function. Global stability: Assume that there exists a scalar function V of the state x, with continuous first order derivatives such that V (x) is positive definite 2

3 V (x) is negative definite V (x) as x then the equilibrium at the origin is globally asymptotically stable. Stability of uniform ultimate boundedness: A solution x, x(t 0 ) = x 0 is said to be uniformly ultimately bounded (UUB) in a hyperball B(0, ɛ) centered at the origin and of radius ɛ, if there exists a non-negative constant Ψ(x 0, B) <, independent of t 0, such that x 0 < δ implies x(t) B for all t t 0 + Ψ(x 0, B). Example: Lyapunov function for LTI systems. Consider the linear system ẋ = Ax, where x R n is the state, A R n n is the system matrix. Propose a quadratic Lyapunov function candidate V (x) = x T P x, where P is a positive definite function to be determined. Taking time derivative yields V (x) = x T ẋ + ẋ T x = x T (A T P + P A)x = x T Qx, where Q is the solution of algebraic Lyapunov equation Q = A T P + P A. Therefore, the system is stable if Q is positive definite or semi-definite. A Lyapunov function successful for stability analysis can be found not by randomly choosing P but only by determining P from the Lyapunov equation for any given positive definite Q. It has been shown that, given a positive definite Q, the system is stable if and only if the unique solution of Lyapunov equation is also positive definite. That is, this backward procedure is necessary and sufficient for both existence of Lyapunov function and analyzing stability. As will be shown later, this systematic way of generating Lyapunov functions for linear systems also applies to many nonlinear (uncertain) systems, for example, the class of feedback linearizable nonlinear systems, the class of nonlinear systems with a linear part, etc. For control design, consider the system ẋ = Ax + Bu, where B R n m is the input matrix, u R n is the input. If the pair (A, B) is controllable, control design and search for Lyapunov function are done through the backward procedure 3

4 as follows: given positive definite matrices Q and R, there is a unique positive definite matrix P satisfying the algebraic Riccati equation A T P + P A P BR 1 B T P Q = 0, then the Lyapunov function is V (x) = x T P x and the stabilizing control is u(x) = R 1 B T P x. The example shows that control design and search of Lyapunov function are integrated and can be done systematically for LTI systems and that Lyapunov functions for linear systems can always be chosen to be quadratic functions. We shall use the above result in chapter three to investigate robust control design for linear and certain nonlinear uncertain systems. Moreover, one of the main objectives of this book is to develop systematic procedures of designing control and searching for Lyapunov function for general nonlinear uncertain systems, though it is not as complete of a solution as the above one for LTI systems. Example: Consider the scalar system given by ẋ = u + a, where a is an uncertain (time-varying) parameter satisfying a < 1. Under the standard linear feedback control law u = kx, the derivative of the Lyapunov function V = 0.5x 2 is given by ( V = kx x a ). k Because of the uncertainty in a, V is only negative definite outside the ball B(0, a/k) B(0, 1/k). Hence, the system is not asymptotically stable, but the solution is given by x = e kt x 0 + a k ( 1 e kt ) a k as t. So, the solution is globally uniformly ultimately bounded (GUUB) with respect to 1/k for the class of uncertainty denoted by a. Furthermore, the bound of GUUB stability tends to the origin as k. The implications of the example are twofold. First, if V is negative definite outside some hyper-ball in state space, stability result of GUUB is concluded. Second, while larger control energy makes the bound of GUUB of the state smaller, no control of finite energy achieves asymptotic stability. Both observations can be extended to general nonlinear systems. The next section addresses robotic manipulator systems which are widely used in the area robust control. Some of the theories developed here are applied to robotic manipulator systems. A brief general discussion is presented below for robotic systems. 4

5 2 Robotic Manipulators A robot is a reprogrammable multifunctional manipulator designed to move material, parts, tools, or specialized devices through variable programmed motions for the performance of a variety of tasks. A robot arm is classified to be either rigid or flexible link. A rigid link could be either revolute (rotary) or linear (prismatic), a prismatic link allows a linear relative motion between any two links, see figure (??). In the chapters to come, all robot manipulator systems discussed are of revolute nature. In the case of a system as complicated as a robot, it is not practical to assume that the parameters in the dynamic model of the robotic system are known precisely. There will always be inexact cancellation of the nonlinearties in the system due to uncertainties. In such cases we use robust control to simplify the equations of motion as much as possible by ignoring certain terms in the equations. One of the uses of robotic systems in the environmental waste management in which accuracy is important specially the accuracy in positioning the end-effector position of the manipulator. Requirement such as safety, motion compliance control, and operation environment can be fulfilled by using low-level robot controller in which the end-effector arm is moved quickly, yet accurately while maintaining a high degree of robustness. Since we are interested in robotic manipulator system as we shall present in chapter 6, let us formulate the dynamical model for a rigid link robot manipulator. The rigid link robot is described by τ = M(q) q + V m (q, q) q + N(q, q) (1) where N(q, q) = G(q) + F ( q) + F M(q) R n n is the inertia matrix, V m (q, q) R n n is a matrix containing the centripetal and Coriolis terms, G(q) R n is the gravity vector, F (q, t) R n is a vector representing lumped uncertainties, q(t) R n is the joint variable vector, and τ R n is the input torque vector. There are three widely used properties of the robot dynamic equation above. These properties will be used in chapter 6, or whenever a robotic system is under study, during the stability analysis of the robust controller. Property 1 5

6 The inertia matrix M(q) is symmetric and positive definite. Hence, m 1 M(q) m 2 (q), where m 1 is a positive constant and m 2 (q) is a strictly positive definite function. Moreover, m 1 and m 2 (q) are chosen in such a way that the maximum possible parameter variation of M(q) is taken into account. Note: For the case that the robotic system is purely revolute, m 2 (q) = m 2 is a positive constant. Property 2 The matrices M(q) and V m (q, q) satisfy the following equation: [ ] 1 x T 2Ṁ(q) V m(q, q) x = 0, x R n. In other words, matrix [ 1 2Ṁ(q) V m(q, q) ] is skew-symmetric. Property 3 The centripetal/coriolis term V m (q, q) is bounded as V m (q, q) a 1 q, and the fiction and gravity terms are bounded as where a i are known constants. G(q) + F d q + F s ( q) a 2 + a 3 q, After introducing the properties used in the analysis of robotic systems, let us discuss briefly a variety of robust control design for robotic systems. Position Control This design technique is used to position a robotic system link(s) to a specific position (desired location) in which accuracy is important especially in industrial and medical robotic systems. Two main types of robust control design schemes have been proposed, one utilizes the so-called Min-Max control and the other uses the saturation type controller. The Min-Max controller is naturally discontinuous and yields global exponential stability, while the saturation controller is continuous but yields global uniform ultimate boundedness. The position control simply drives the robotic link(s) to a final desired position with a very small error, which is referred to as a set point tracking. 6

7 Force Control Many control design schemes have been developed for robotic systems in free space. This is, a robot arm is not in contact with any surface. However, most industrial robots used for yelding, grinding, polishing, etc.., require contact with objects or surfaces. Hence, the robot arm motion is constrained depending on the direction of the arm movement. This fact motivated researchers to investigate the constrained motion case and develop position/force controllers. Among these controllers are hybrid position/force control, impedance control, and reduced order methods. The disadvantage of the hybrid position/force control is that it requires exact knowledge of the robot manipulator and thus, the analysis is limited to the uncertainty free systems. An adaptive control design scheme was developed for hybrid position/force robots with uncertainty which is based on the joint-space robot model formulation. Impedance Control Impedance control is based on the idea that the robust controller should be utilized to regulate the dynamic behavior between the robot arm end-effector motion and the force exerted on the surface, rather than considering the motion and force control problems separately. The name impedance emanates from the idea of using an Ohm s law type relationship between motion and force. Similar to previous types of controllers, impedance controller has been extensively studied. A robust impedance controller was developed to ensure stability in lieu of uncertainties. An adaptive impedance controller was also developed that takes care of parametric uncertainty. Industrial Robots In present days, adaptive control is widely utilized in industrial robots because of the advantage of the inexpensive computer power that has become available. Moreover, these robots are being utilized to their full potential in terms of the speed and precision of their movements. It is possible to use a dynamic model of the manipulator as the heart of the sophisticated control algorithm with a powerful control computer. This dynamic model allows the control algorithm to know how to control the manipulator s actuators in order to compensate for the complicated effects of inertia, centripetal, Coriolis, gravity, and friction forces when the robot is in motion. The result is that the manipulator can be made to follow a desired trajectory through space with smaller tracking errors. Adaptive control, as 7

8 other types of controllers, has its advantages and disadvantages. Adaptive control cannot be utilized to estimate system with fast time-varying uncertainties or parameters because one cannot predict the nature of the uncertainty and the adaptive algorithm may not be able to adapt to fast enough to the time-varying parameters. On the other hand, a robust controller, used mostly in this dissertation, can stabilize nonlinear systems with arbitrary fast time-varying uncertainties or parameters. Moreover, we shall introduce robust control design techniques for robotic systems with arbitrary fast time-varying uncertainties since robust control design requires only known bounding functions of the uncertainties. This dissertation focuses on nonlinear robust control design schemes. 3 Robust control design under Matching Conditions Many primary results of nonlinear uncertain systems under matching conditions have been developed in the last 15 years. Gutman introduced a discontinuous min-max control which yields asymptotic stability for nonlinear systems under the matching condition. Because of the discontinuity behavior of the controller, it is physically poorly behaved since all physical systems have a finite bandwidth, but the discontinuous control requires systems with infinite bandwidth. Later, Corless and Leitmann introduced a class of continuous state feedback controller guaranteeing uniform ultimate boundedness under the matching conditions. The mathematical model of nonlinear uncertain systems under matching conditions is established through the following definition. Definition: Consider the following nonlinear uncertain system ẋ = f(x, t) + f(x, t) + B(x, t)u + B(x, t)u (2) where f(x, t) and B(x, t) are the unknown parts of f(x, t) and B(x, t), respectively. The system is said to satisfy the matching conditions MCs if uncertainty f(x, t) can be decomposed as f(x, t) = B(x, t) f (x, t), B(x, t) = B(x, t) B (x, t), and if there exists a positive constant ɛ such that, B (x, t) 1 ɛ. (3) Therefore, the system can be rewritten as ẋ = f(x, t) + B(x, t) [ f (x 1 ) + (1 + B (x, t)) u(x, t)] (4) 8

9 in which the uncertainty enters the system through the same channel as control input u. The reason behind inequality (3) is twofold, first, the system is not stabilizable for the case when B (x, t) = 1. Moreover, if B (x, t) > 1, then term 1+ B (x, t) is uncertain and hence, any input control may cause the state to grow out of bound. Second, the inequality ensures that there is no singularity in the control design by guaranteeing that term 1 + B (x, t) is invertible. Remark: If the uncertainty would be known, one can easily choose a control input to cancel its effect and achieve stability. But, since physical dynamical systems contain some uncertainties which are unknown, one replaces those uncertainties by their bounding functions which are chosen depending of the structure of the system and then the robust control design scheme can be adopted. we shall investigate system stability through Lyapunov s direct method. 3.1 Lyapunov stability in Robust Control Design The nominal model of system (4) is given by ẋ = f(x, t) + B(x, t)u(x, t) (5) We shall assume that the origin (x = 0) is globally asymptotically stable for the uncontrolled system ẋ = f(x, t). Furthermore, suppose that there exists a Lyapunov function for system (5), i.e., there exists a continuously differentiable function V (x, t) that satisfies the following inequalities, for all (x, t) [0, ) δ 1 ( x ) V (x, t) δ 2 ( x ), V t + V x [f(x, t)] δ( x ) (6) where δ i are class K functions. To demonstrate the stability of system (4), choose input control u(x, t) to be of the form µ(x, t) u(x, t) = ρ(x, t), (7) ɛ( µ(x, t) + εϕ(t) where ε > 0 and ϕ(t) an L 1 function, are chosen freely by the designer and µ(x, t) = B(x, t) V ρ(x, t) x f (x, t) ρ(x, t). 9

10 Differentiating V (x, t) under robust control (7) yields V = V t + V x [f(x, t) + B f + B (1 + B ) u] δ( x ) + V x [B f + B (1 + B ) u] V δ( x ) + x B ρ(x, t) + V x B (1 + B ) u δ( x ) + µ µ2 (x, t) (1 + B ) ɛ( µ(x, t) + εϕ(t) µ 2 (x, t) δ( x ) + µ ( µ(x, t) + εϕ(t) ε µ(x, t) ϕ(t) δ( x ) + ( µ(x, t) + εϕ(t) δ( x ) + εϕ(t) (8) The following results are deduced from robust control design under matching conditions. 1. If ϕ(t) is constant, say ϕ(t) = 1, then the system is globally uniformly ultimately bounded with an ultimate bound given by a class K function of ɛ over infinite time horizon. 2. If ϕ(t) is an exponentially decaying function, say ϕ(t) = e at, for some a > 0, then the system is globally exponentially stable. In summary, one can apply the above systematic design scheme to systems satisfying the matching conditions. The mechanical dynamics of a rigid-link robotic manipulator for instance, is an example of a physical system satisfying the matching conditions. However, there are many uncertain nonlinear systems that do not satisfy the matching conditions. The next section introduces robust control design scheme for systems satisfying the so-called equivalently matched uncertainty. 3.2 Examples of Unstabilizable Uncertain Systems Although it would be ideal that robust control can be designed to stabilize all uncertain systems in the form of (??), the following examples show that not all uncertain systems are stabilizable. Example: Consider the second-order system ẋ 1 = x 2 + (x 1, x 2 ), ẋ 2 = u, 10

11 in which the uncertainty ( ) is bounded as (x 1, x 2 ) 2+x 2 1 +x 2 2. One can easily see that the system with any admissible uncertainty is not stabilizable since a possibility of additive uncertainty 1 (x 1, x 2 ) within the given bounding function is x 2 + x 1. The system is not stabilizable since uncertainty within its bound can change the structure of the system such that part of system dynamics becomes unstable and decoupled from the rest of the system and from control input. Example: Consider the scalar system ẋ = x + [1 + (x)]u, where uncertainty is bounded as (x) C for some C 1. The system is not stabilizable since (x) could be 1, and then the system is not subject to any control. The uncertainty (x) may be such that 1 + (x) is uncertain because of C > 1, and therefore any control introduced may have adverse effect since it may cause the state to grow out of bound more quickly. In fact, whenever there is a large multiplicative uncertainty associated with the control input, no control is the best choice, and the uncertain system becomes unstabilizable if any control is needed. It is worth noting that the first subsystem in Example?? becomes this example if (x 1, x 2 ) = x 1 + (x 1 )x 2. Example: Consider the scalar system ẋ = (x) + u 2, where uncertainty is bounded as (x) 1. The system is not stabilizable since, no matter what choice is made for u, the control action in ẋ is always unidirectional (positive). In fact, any scalar uncertain system is not stabilizable if the designer cannot make ẋ be both positive and negative upon his choice through selecting u (specifically, through choosing robust control to dominate all possible uncertainties). Example: Consider the system ẋ 1 = 11 x 1 + x x 3 ẋ 2 = 21 x x 2 + x 3 ẋ 3 = u, where uncertain terms ij are independent but bounded by constants C ij > 0. The system is not stabilizable for many sets of constants C ij. To see this conclusion, consider the simplest case that the uncertainties are time-invariant and state-independent. In this case, 11

12 the transfer function between u and x 1 is and the controllability matrix is X 1 (s) U(s) = 13 (s 22 ) + 1 s(s 11 )(s 22 ) 21 s, C = The zero z of the transfer function and the determinant of controllability matrix are, respectively, z = , and det(c) = If det(c) = 0, the system becomes uncontrollable due to pole-zero cancellation, and the cancellation may occur in the right half of the s plane. Uncontrollability due to unstable pole-zero cancellation implies that the system cannot be stabilized. For the system under consideration, the presence of uncertainty 13 of potentially large size implies that this kind of instabilizability may arise unless certain size limitations in terms of the bound of 13 are imposed on the maximum magnitudes of 11, 21 and 22. Relationship between bounding functions of uncertainties can be found through robust control design to guarantee both stabilizability and robust stability. There are many other uncertain systems in which unstable, uncontrollable pole-zero cancellation may occur. Although dynamics of the above examples are simple, they show existence of unstabilizable systems and, more importantly, provide intuitive explanations of what may cause systems to be unstabilizable. Specifically, there are two categories in the state space: loss of controllability and control contribution to differential equation being either unknown or only unidirectional (as shown in second and third examples). In first and last examples, the two systems have isolated subsystem or pole-zero cancellation and therefore are uncontrollable. As a result of the above examples, it is crucial to identify stabilizable uncertain systems and to design robust control for those systems. Robust control theory is to identify the class of all stabilizable uncertain systems and to provide stabilizing controls that guarantee desired performance. The ultimate objective of robust control theory of nonlinear uncertain system is twofold. First, if necessary, determine the least requirements, called structural conditions, on the system (either in terms of system structure or location of uncertainty) such that it can be stabilized or controlled. Second, find procedures under which robust control u can be 12.

13 systematically designed. The key issue in the design is the search of Lyapunov functions and their associated robust controllers (which may be different for achieving various types of performances). 4 Back-Stepping Design Procedure The backstepping design procedure can be seen from the following simple example. Example: Consider the second-order system: ẋ 1 = x 2, ẋ 2 = u. This system is linear and consists of two cascaded integrators. A linear stabilizing control can be designed by solving a simple Lyapunov equation. The Riccati equation can be used to design robust control if there are linearly bounded uncertainties. However, those procedures do not apply to nonlinear systems since they depend on linear matrix equations. Here, we plan to start an intuitive design that can be extended later to nonlinear systems. From the second equation, we see that u can control x 2 to anywhere. For the first equation, if x 2 were a control variable, an obvious stabilizing control would be x 2 = x 1. Since x 2 is not a control but a state variable, the equation x 2 = x 1 does not make any sense. To distinguish the state variable x 2 and the control designed for x 2 from the actual control u, let us call the control designed for x 2 fictitious control and denote it by x d 2 = x 1. Although the fictitious control is not implementable, we can rewrite the first equation as ẋ 1 = x 1 + (x 2 + x 1 ) = x 1 + (x 2 x d 2). This simple manipulation reveals intuitively that stabilization of the first equation may be achieved if we can make x 2 x d 2 = x 2 + x 1 converge to zero. Hence, fictitious control x d 2 can be viewed as the desired trajectory for state variable x 2. Recall that, in the second equation, control u can be designed to drive x 2 anywhere. The problem of making x 2 track x d 2 is equivalent to making the new, translated state variable z 2 = x 2 x d 2 converge zero (that is, a stabilization problem). The dynamics of z can be found as follows: ż 2 = ẋ 2 ẋ d 2 = ẋ 2 + ẋ 1 = u + x 2. Obviously, the control u = x 2 z 2 = x 2 (x 2 + x 1 ) guarantees asymptotic stability of z 2. Once z 2 = x 2 + x 1 converges to zero, x 1 will approach zero by the design of x d 2 in 13

14 ẋ 1 = x 1 + z 2 (which is stable if z 2 = 0), and consequently x 2 goes to zero. Therefore, the overall system is asymptotically stable. This intuitive argument of stability can be verified by a simple Lyapunov proof. Choosing Lyapunov function V = x z2, 2 one can easily show that the control u = x 2 (x 1 + x 2 ) yields global asymptotic stability. In fact, the Lyapunov function is the sum of Lyapunov functions for subsystems of the states x 1 and z 2. The control in this example is designed by working sequentially through two integrators. In the process, a fictitious control is design, a state transformation is performed in which fictitious control is differentiated. Such a design is called a recursive design since, by the transformation, the design of fictitious control is imbedded into the actual control design. The design is also called backstepping or backward recursive because the direction in which the sequential design is proceeded is the opposite to the direction of signal flow graph of the system, that is, the direction at which physical information flows within the system. This approach which obviously works systematically for multiple-integrator systems was realized in the sixties. But applications of its extensions to nonlinear control, adaptive control, and robust control have been developed only in past several years. Mathematically, the design procedure can be genearlized and applied to nonlinear systems because of the following reasons. First, by introducing a fictitious control variable to a given subsystem, its dynamics satisfy locally the matching conditions with respect to the fictitious control and therefore can be compensated. Second, state transformation make the difference between dynamics of fictitious control and its corresponding state variable equivalently matched and therefore can be compensated. Finally, sub-lyapunov functions can be easily found for all subsystems since they are of first order, and the overall Lyapunov function is simply the sum of sub-lyapunov functions, by which stability of the overall system can be concluded. 14