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Dr. Javed Iftikhar delivered this lecture at A.P. University of Law for Non Linear Control Systems course. It includes: Feedback, Linearization, Internal, Dynamics, Systems, Diffeomorphism, Frobenius, Theorem, Relative, Degree
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Department of Electrical Engineering
Fall 2009
Feedback Linearzation Input-State Linearization Input-Output Linearization Internal Dynamics of Linear Systems Zero-Dynamics Preliminary Mathematics Diffeomorphism Frobenius Theorem Input-State Linearization Control Design Input-Output Linearization Well Defined Relative Degree Undefined Relative Degree Normal Form Zero-Dynamics Local Asymptotic Stabilization Global Asymptotic Stabilization
I (^) The dynamics:
A(h) ˙h(t) = u − a
2 gh
where A(h) is the cross section of the tank and a is the cross section of the outlet pipe. I (^) Choose u = a
2 gh + A(h)v h˙ = v I (^) Choose the equivalent input v : v = −α˜h where ˜h = h(t) − hd is error level, α a pos. const. I (^) ∴ resulting closed-loop dynamics: ˙h + α˜h = 0 ⇒ h˜ → 0 as t → ∞ I (^) The actual input flow: u = a
2 gh + A(h)α˜h I (^) First term provides output flow a√ 2 gh I (^) Second term raises the fluid level according to the desired linear dynamics I (^) If hd is time-varying: v = ˙hd (t) − α˜h
I (^) Canceling the nonlinearities and imposing a desired linear dynamics, can be simply applied to a class of nonlinear systems, so-called companion form, or controllability canonical form: I (^) A system in companion form:
x(n)(t) = f (x) − b(x)u (1)
I (^) u is the scalar control input I (^) x is the scalar output;x = [x, x˙, ..., x(n−1)] is the state vector. I (^) f (x) and b(x) are nonlinear functions of the states. I (^) no derivative of input u presents.
I (^) (1) can be presented as controllability canonical form
d dt
x 1 .. . xn− 1 xn
x 2 .. . xn f (x) + b(x)u
I (^) for nonzero b, define control input: u = (^1) b [v − f ]
I (^) A two-link robot: each joint equipped with I (^) a motor for providing input torque I (^) an encoder for measuring joint position I (^) a tachometer for measuring joint velocity I (^) objective: the joint positions ql and q 2 follow desired position histories qdl (t) and qd 2 (t) I (^) For example when a robot manipulator is required to move along a specified path, e.g., to draw circles.
I (^) Using the Lagrangian equations the robotic dynamics are:[ H 11 H 12 H 21 H 22
] [ q¨ 1 q ¨ 2
]
[ −h q˙ 2 −h q˙ 2 − h q˙ 1 h q˙ 1 0
] [ q˙ 1 q ˙ 2
]
[ g 1 g 2
[ τ 1 τ 2
]
where q = [q 1 q 2 ]T^ : the two joint angles, τ = [τ 1 τ 2 ]T^ : the joint inputs, and H 11 = m 1 l c^21 + l 1 + m 2 [l^21 + l c^22 + 2l 1 lc 2 cos q 2 ] + I 2 H 22 = m 2 l c^22 + I 2 H 12 = H 21 = m 2 l 1 lc 2 cos q 2 + m 2 l c^22 + I 2 g 1 = m 1 lc 1 cos q 1 + m 2 g [lc 2 cos(q 1 + q 2 ) + l 1 cos q 1 ] g 2 = m 2 lc 2 g cos(q 1 + q 2 ), h = m 2 l 1 lc 2 sin q 2
I (^) Control law for tracking, (computed torque): [ τ 1 τ 2
[ H 11 H 12 H 21 H 22
] [ v 1 v 2
]
[ −h q˙ 2 −h q˙ 2 − h q˙ 1 h q˙ 1 0
] [ q˙ 1 q ˙ 2
]
[ g 1 g 2
]
where v = ¨qd − 2 λ˜q˙ − λ^2 q˜, ˜q = q − qd : position tracking error, λ: pos. const. I (^) ∴ ˜¨qd + 2λ˜q˙ + λ^2 ˜q = 0 where ˜q converge to zero exponentially.
I (^) Consider (^) x˙ 1 = − 2 x 1 + ax 2 + sin x 1 x ˙ 2 = −x 2 cos x 1 + u cos(2x 1 ) I (^) Equ. pt. (0, 0) I (^) The nonlinearity cannot be directly canceled by the control input u I (^) Define a new set of variables: z 1 = x 1 z 2 = ax 2 + sin x 1 ∴ z˙ 1 = − 2 z 1 + z 2 z ˙ 2 = − 2 z 1 cos z 1 + cos z 1 sin z 1 + au cos(2z 1 ) I (^) The Equ. pt. is still (0, 0). I (^) The control law: u = (^) a cos(2^1 z 1 ) (v − cos z 1 sin z 1 + 2z 1 cos z 1 ) I (^) The new dynamics is linear and controllable: ˙z 1 = − 2 z 1 + z 2 , z˙ 2 = v I (^) By proper choice of feedback gains k 1 and k 2 in v = −k 1 z 1 − k 2 z 2 , place the poles properly.
I (^) The result is not global. I (^) The result is not valid when xl = (π/ 4 ± kπ/2), k = 0, 1 , 2 , ...
I (^) The input-state linearization is achieved by a combination of a state transformation and an input transformation with state feedback used in both. I (^) To implement the control law, the new states (z 1 , z 2 ) must be available. I (^) If they are not physically meaningful or measurable, they should be computed by measurable original state x. I (^) If there is uncertainty in the model, e.g., on a error in the computation of new state z as well as control input u. I (^) For tracking control, the desired motion needs to be expressed in terms of the new state vector. I (^) Two questions arise for more generalizations which will be answered in next lectures: I (^) What classes of nonlinear systems can be transformed into linear systems? I (^) How to find the proper transformations for those which can?
I (^) Consider (^) x˙ 1 = sin x 2 + (x 2 + 1)x 3 x ˙ 2 = x 15 + x 3 x ˙ 3 = x 12 + u y = x 1
I (^) To generate a direct relationship between the output y and the input u, differentiate the output ˙y = ˙x 1 = sin x 2 + (x 2 + 1)x 3 I (^) No direct relationship differentiate again: ¨y = (x 2 + 1)u + f (x), where f (x) = (x 15 + x 3 )(x 3 + cos x 2 ) + (x 2 + 1)x 12 I (^) Control input law: u = (^) x 21 +1 (v − f ). I (^) Choose v = ¨yd − k 1 e − k 2 e˙, where e = y − yd is tracking error, k 1 and k 2 are pos. const. I (^) The closed-loop system: ¨e + k 2 e˙ + k 1 e = 0
I (^) The control law is defined everywhere except at singularity points s.t. x 2 = − 1 I (^) To implement the control law, full state measurement is necessary, because the computations of both the derivative y and the input transformation need the value of x. I (^) If the output of a system should be differentiated r times to generate an explicit relation between y and u, the system is said to have relative degree r. I (^) For linear systems this terminology expressed as # poles −# zeros. I (^) For any controllable system of order n, by taking at most n differentiations, the control input will appear to any output, i.e., r ≤ n. I (^) If the control input never appears after more than n differentiations, the system would not be controllable.
I (^) I-O linearization can also be applied to stabilization (yd (t) ≡ 0): I (^) For previous example the objective will be y and ˙y will be driven to zero and stable internal dynamics guarantee stability of the whole system. I (^) No restriction to choose physically meaningful h(x) in y = h(x) I (^) Different choices of output function leads to different internal dynamics which some of them may be unstable. I (^) When the relative degree of a system is the same as its order: I (^) There is no internal dynamics I (^) The problem will be input-state linearization
I (^) Feedback linearization cancels the nonlinearities in a nonlinear system s.t. the closed-loop dynamics is in a linear form. I (^) Canceling the nonlinearities and imposing a desired linear dynamics, can be applied to a class of nonlinear systems, named companion form, or controllability canonical form. I (^) When the nonlinear dynamics is not in a controllability canonical form, input-state linearization technique is employed:
I (^) Now consider a little different dynamics
[ x˙ 1 x ˙ 2
x 2 + u −u
y = x 1
I (^) using the same control law yields the following internal dynamics
x˙ 2 − x 2 = e(t) − y˙d
I (^) Although yd and y are bounded, x 2 and u diverge to ∞ as t → ∞ I (^) why the same tracking design method yields different results? I (^) Transfer function of (3) is: W 1 (s) = s+1 s 2. I (^) Transfer function of (4) is: W 2 (s) = s− s 21. I (^) ∴ Both have the same poles but different zeros I (^) The system W 1 which is minimum-phase tracks the desired trajectory perfectly. I (^) The system W 2 which is nonminimum-phase requires infinite effort for
I (^) Consider a third-order linear system with one zero
x˙ = Ax + bu, y = cT^ x (5)
I (^) Its transfer function is: y = (^) a b^0 +b^1 s 0 +a 1 s+a 2 s^2 +a 3 s^3 u I (^) First transform it into the companion form:
z ˙ 1 z ˙ 2 z ˙ 3
−a 0 −a 1 −a 2
z 1 z 2 z 3
(^) u (6)
y = [b 0 b 1 0]
z 1 z 2 z 3
