Lyapunov Based Control
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1 Lyapunov Based Control Control Lyapunov Functions Consider the system: x = f(x, u), x R n f(0,0) = 0 Idea: Construct a stabilizing controller in steps: 1. Choose a differentiable function V: R n R, such that V(x) > 0, x 0, V(0) = 0 (1). For all x R n, choose u = u(x), such that V(x). f(x, u) < 0 Stability follows from Lyapunov s direct method for the autonomous system x = f(x, u(x)) Definition: A function V( ) for which equation (1) holds, and for which u V(x). f(x, u) < 0, x Is called a control Lyapunov function (clf). Example Consider the system x 1 = x 1 + x φ(x 1, x ) x = ψ(x 1, x ) + u With φ(, ), ψ(, ) continuous, u R. Choose V(x 1, x ) = 1 (x 1 + x ), then u R V(x). f(x, u) = u R { x x 1 x φ(x 1. x ) + x u + x ψ(x 1, x ) } = x 0 x 1 4 x = 0 < 0 Thus V(x) is a clf for the system, and the system is globally asymptotically stabilizable. Choosing Yields u = x φ(x 1, x ) ψ(x 1, x )x 1 V. f(x, u) = x 1 4 x < 0 1
2 Remarks 1. The idea may be generalized to stabilize systems with disturbances, or with general uncertainties. The method is to bound the uncertainties, and choose a control which provides a sufficient margin to guarantee negativity of the Lyapunov function for the closed loop system.. Existence of a clf is necessary and sufficient for stabilizability of the system.. This method does not guarantee performance. It establishes only stability of the closed loop system. References 1. Sastry. Freeman & Kokotovich, Robust Nonlinear Control. Main technique: inverse optimality (outside scope of this course). Sepulchre, Jankovic & Kokotovic, Constructive Nonlinear Control. Main technique: Passivity and Disipativity. Stabilization via Backstepping Reference: According to Sastry pp The stabilization technique Backstepping relies on a specific structure of the nonlinear system. Before applying it, the system must be re-formed to conform to this structure. Consider an n-dimensional nonlinear system of the form: x 1 = x + f 1 (x 1 ) x = x + f (x 1, x ) x i = x i+1 + f i (x 1, x,, x i ) x n = f(x 1, x,, x n ) + u Note: The i th derivative x i depends on the first to i th states - x 1, x,, x i nonlinearly and is affinely dependant on the next state, x i+1, while the final state depends affinely on u. Here the block diagram for n =.
3 From this block diagram it becomes evident that: With u it is possible to directly control x. With x it is possible to directly control x. With x it is possible to directly control x 1. This allows development of a strategy to solve the stabilization problem. Proceeding formally, we start with x 1 and recursively define a change of coordinates together with a Lyapunov function that simplifies the system and provides a controller which then stabilizes the system. Define: z 1 = x 1 z = x α 1 (x 1 ) α 1 (x 1 ) x 1 f 1 (x 1 ) And introduce the partial Lyapunov function V 1 (z 1 ) = 1 z 1. It then follows that z 1 = z 1 + z z = x + f (x 1, x ) α 1 (x 1 ). (x + f 1 (x 1 )) x + f (z 1, z ) And that V 1 = z 1 + z 1 z Proceeding recursively: z = x α (z 1, z ) V = V z Substituting into the expression for z : z = z + α (z 1, z ) + f (z 1, z ) Choosing α z 1 z f (z 1, z ), it follows that At the i th step, we choose z 1 = z 1 + z z = z 1 z + z V = z 1 z + z 1 z z i = x i+1 α i (z 1,, z i ) V i = V i z i = 1 z j i j=1 With the appropriate choice of α_i, it follows that z i = z i 1 z i + z i+1 V i = z 1 z z i + z i z i+1 To complete the operation we choose the control
4 u = α n (z 1,, z n ) So that z n = z n 1 z n V n = z 1 z z n Remarks 1. Note that we have defined a coordinate transformation and control so that the final system a. Has stability proven via the Lyapunov function V n b. Has linear dynamics. Applying this recipe blindly leads to very complex α i. It is possible to choose other Lyapunov functions and other α i. In this case it is important to choose so that the triangular structure is maintained, and stability is chosen. i.e. V i is chosen so that And α i is chosen so that V i = F i (z 1, z,, z i+1 ) F(z 1, z,, z i, 0 ) < 0 Example: Apply the backstepping algorithm to stabilize the system: x 1 = x x 1 x = u First define z 1 = x 1 z = x α 1 (x 1 ) V 1 = 1 z 1 z 1 = x 1 = x x 1 Variant 1 (standard algorithm): α 1 (x) = x 1 + x 1 Then z 1 = z 1 + z V 1 = z 1 + z 1 z z = x α 1 x 1 = u (x 1 1)(x x 1 ) = u (z 1 1)(z + z 1 z 1 z 1 ) = u (z 1 1)(z z 1 ) Choosing u as follows 4
5 u = z 1 z + (z 1 1)(z z 1 ) Yields the system z 1 = z 1 + z z = z 1 z V = z 1 z Variant (simplified version): ): α 1 (x) = 0 z 1 = x 1 z = x Then the Lyapunov function and derivative are given by V 1 = 1 z 1 z 1 = x 1 = z z 1 Choosing now V 1 = z z 1 z z = u u = z 1 z z 1 = z z 1 z = z 1 z V = 1 (z 1 + z ) V = z z 1 z z 1 z z = z 1 4 z Note that the result is the same: A provably stable system. But the algebra is significantly simpler Exercise: Design a stabilizing controller for the following system via backstepping. x 1 = x 1 + x x = x + x x = u 5
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