Review of course Nonlinear control Lecture 5. Lyapunov based design Torkel Glad Lyapunov design Control Lyapunov functions Control Lyapunov function

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1 Review of corse Nonlinear control Lectre 5 Lyapnov based design Reglerteknik, ISY, Linköpings Universitet Geometric control theory inpt-otpt linearization controller canonical form observer canonical form Lyapnov theory Stability reslts Passivity Circle and Popov criteria Lyapnov design Control Lyapnov fnctions Control Lyapnov fnctions Back-stepping Forwarding Observers 5 (Distrbance spression) 6 (Passivity based control) V is a Control Lyapnov fnction if for every x there is some so that V x f (x, ) < Choose = k(x) so that V x f (x, k(x)) negative definite Might be difficlt to find nice k

2 Control Lyapnov fnction Example A typical block strctre x m k The spring k has cbic stiffness ẋ = x Actator act System ẋ = x + x Possible control Lyapnov fnctions: z x V = x + }{{} potential energy x }{{} kinetic energy, V = x + x + x Design a controller assming act is the control signal Step back to the real control signal and extend the controller design ( backstepping ) Back-stepping with Lyapnov fnctions Repeated backstepping If the actator is first order one can take act = z: ẋ = f (x) + g(x)z ż = a(x, z) + b(x, z) Sppose we find a control law z = k(x) and a Lyapnov fnction V(x) so that V x (x)(f (x) + g(x)k(x)) = W(x), V, W positive definite This control law can then be extended (the step back ) to a control law for, sing a control Lyapnov fnction, eg Systems in feedback form: ẋ = f (x ) + g (x )x ẋ = f (x, x ) + g (x, x )x ẋ n = f n (x,, x n ) + g n (x,, x n ) ẋ = f (x, x ) ẋ = f (x, x, x ) ẋ n = f n (x,, x n, ) Repeated backstepping is easily done for the strctre to the left It can also be generalized to the strctre to the right V e (x, z) = V(x) + (z k(x))

3 x z Back-stepping Forwarding Extending the Lyapnov fnction when an integrator is added to the otpt Basic advantage: Not necessary to cancel terms that make V e negative Many opportnities for creative extensions ẋ = f(x)+g(x) y z y = a(x)+b(x) s Ttorial, theory and applications in: Ola Härkegård: Back-stepping and Control Allocation with Applications to Flight Control PhD Thesis, Department of Electrical Engineering, Linköping University, Sppose positive definite fnctions V, W and a control law k are known so that V x (x)(f (x) + g(x)k(x)) = W(x) Let ζ = φ(x, z) be constant when = k(x) Then V e (x, z) = V(x) + ζ is a sitable control Lyapnov fnction Forwarding example Reslt, forwarding example Forwarding control for: x() = 65, z() = ż = x + ẋ = x + Stabilization of x-system: = k(x) = Coordinate change:ζ = z + x Reslting control law: = x ( + x)ζ t t Left diagram: x, right diagram: z fll forwarding controller: red linear part of the controller: ble

4 The high gain observer A Lyapnov fnction for the observer ẋ = Ax + Bφ(x) + g(x), y = Cx A =, B = C = [ ] The observer is ˆx = Aˆx + Bφ(ˆx) + g(ˆx) + K(y Cˆx) The observer error is x = (A KC) x + B(φ(x) φ(ˆx)) + (g(x) g(ˆx)) }{{} L(x,ˆx,) With K = S C T and S given by A T S + SA C T C = θs the fnction V = x T S x is a Lyapnov fnction, if θ is large enogh State feedback via observer Distrbances State feedback: ẋ = f (x) + g(x), = k(x) Lyapnov fnction V: V = V x (x)(f (x) + g(x)k(x)) q(x) x w Observer: observer error x = x ˆx Lyapnov fnction V e = x Q for some norm Q: V e = q e (x) Is W(x, x) = V(x) + V e ( x) a Lyapnov fnction for the closed loop system? ẋ = x x ẋ = x x x + + w

5 Distrbance sppression Distrbance example Reslt of distrbance compensation To the left: V(t), to the right: x (t) Uncontrolled system withot distrbance: dotted, ncontrolled system with distrbance: dashed, controlled system with distrbance: solid Distrbance w, control signal, and filtered control signal Passivity Passivity and feedback Passive system: T T y dt + γ(x()) Design a feedback law = v k(x) If two passive systems are connected in a feedback loop, the reslting system is passive: + S y y so that the system is still passive from v to y y S