Mathematics for Control Theory Geometric Concepts in Control Involutivity and Frobenius Theorem Exact Linearization Hanz Richter Mechanical Engineering Department Cleveland State University
Reading materials Reference: Harry Kwatny and Gilmer Blankenship [2000], Nonlinear Control and Analytical Mechanics: A Computational Approach, Birkhäuser, ISBN 978-0-8176-4147-4, Sections 3.3.4 and 3.3.5. www.pages.drexel.edu/~hgk22/courses/mem636_638/ncam.pdf Jean-Jacques Slotine and Weiping Li [1991], Applied Nonlinear Control, Prentice-Hall, ISBN: 0-13-040890-5, Chapter 6. 2 / 11
More insight from car parking example Given a point (x p,y p,φ,θ), and two instantaneous values of control u 1 and u 2, the four differential equations describing cart dynamics provide a tangent vector ẋ (an instantaneous direction of motion in 4 coordinates). We see that our choices of instantaneous values of u 1 and u 2 can be used to point the cart in various directions. Pointwise, the tangent vector is a simple linear combination of d(x(t)) and s(x(t)), with u 1 and u 2 as scalar weights. But our problem is not merely to point the cart. We want to reach new points by manipulating u 1 and u 2 continuously. A simple subclass of control actions is given by piecewise constant functions of time, with the restriction that u 1 u 2 = 0 (follow the direction of only one vector field at a time). 3 / 11
Insight from car parking example... If a control action (u 1,u 2 ) with u 1 u 2 = 0 which is constant in a time interval is used, the flow of the differential equation will transfer the initial point to a new one. At this new point, we can form a new linear combination to point the tangent vector in a desired direction. Clearly, an understanding of achievable directions and achievable trajectories is necessary. We used a sequence of forward and backward motions along the flows generated by two vector fields: K(t,x) = Φ t g Φ t f Φ t g Φ t f x The operator K(t,x) yielding the final point starting from x is called the commutator of f and g. 4 / 11
Commutator and Lie Bracket We saw that K(t,x)x = x iff [f,g] = 0. There s a more precise relationship (See Kwatny and Blankenship, Proposition 3.67): K(t,x) defines a continuous path in M and: d dt K(0+,x) = [f,g] x Examine the proof and pay close attention to Fig. 3.7. 5 / 11
Control by Exact Linearization Input to state linearization, also called exact linearization, if possible, provides a simpler way to specify control actions. For instance, a 3rd-order feedback linearizable system ẋ = f(x)+g(x)u can be written as: ż 1 = z 2 ż 2 = z 3 ż f = v where u = u(z,v) is a suitable input transformation and v is a new control. Then control design becomes trivial using v as the control. The state and control transformations are use to express the control law in original coordinates. 6 / 11
Complete Integrability Refer to Slotine and Li, Sect. 6.2 for the motivation of the definition below. Suppose the set {f i (x)},i = 1,2...m is linearly independent in some subset U R n. The set is said to be completely integrable in U if we can find n m scalar functions h : R n R satisfying: h i f j = 0 for 1 i n m, 1 j m, with the set of { h i (x)} linearly independent in U. Under this definition, we have m(n m) partial differential equations. If there s a solution, the set of vector fields is completely integrable. Exercise: Determine if the following set is completely integrable (can you find an subset of R n where the vector fields are linearly independent?) f(x) = [sin(x 1 ) 0 cos(x 2 )] T, g(x) = [cos(x 3 ) x 2 sin(x 2 )] T 7 / 11
Involutivity and Frobenius Theorem Suppose that for any pair of vector fields f i,f j from a linearly independent set {f 1,f 2,...f m }, the Lie bracket [f i,f j ] can always be expressed as a linear combination of the vector fields in the set (with possibly state-dependent coefficients). Then the set is involutive. The test for involutivity is just a pointwise rank test: for any i,j, [f i (x),f j (x)] must already be in the span of {f 1 (x),f 2 (x),...f m (x)} for all x in the subset where involutivity is to be verified. Example: Use a numerical or analytical procedure to test involutivity of the set {f,g} of the previous example in the same region where these fields are linearly independent. Frobenius Theorem: The linearly independent set {f 1,f 2,...f m } is completely integrable if anf only if it is involutive. 8 / 11
Exact Linearization of Single-Input Affine Systems Consider the system ẋ = f(x)+g(x)u where f and g are smooth vector fields. The system is input-to-state, or exactly linearizable if there is a diffeomorphic state transformation Φ : Ω R n with z = Φ(x) and an input transformation u = α(x)+β(x)v in some region Ω R n such that the new state z and the new input v are related by the linear, time-invariant state equation below: ż 1 = z 2 ż 2 = z 3. ż n = v Note that there s no output defined, and if this ambitious transformation is possible, there are no internal dynamics left. Among mechanical systems, robot dynamics are an important case of input-state linearizable systems with many inputs. 9 / 11
Necessary and Sufficient Condition for Exact Linearizability (Theorem 6.2 in Slotine and Li): The single-input, n-state nonlinear system ẋ = f(x)+g(x)u is exactly linearizable iff the following conditions hold in some region Ω R n : 1. The vector fields {g,ad f g,...ad f n 1g} are linearly independent in Ω 2. The set {g,ad f g,...ad f n 2g} is involutive in Ω. What happens with these conditions for linear systems (f(x) = Ax and g(x) = B)? The proof of this Theorem includes a method to construct the diffeomorphic state transformation and the input transformation. Suppose the system is exactly linearizable and let z(x) = [z 1 (x) z 2 (x)...z n (x)] T. Find z 1 first, to satisfy z 1 ad f ig = 0, i = 1,2..n 2 z 1 ad f n 1g 0 10 / 11
Recipe for Exact Linearizability... The state transformation is z(x) = [z 1 L f z 1...L f n 1z 1 ] T while the input transformation uses α(x) = Ln f z 1 L g L f n 1z 1 1 β(x) = L g L f n 1z 1 Examples: We solve Problems 6.3 and 6.7 from Slotine and Li. 11 / 11