Contents lecture 6 (7) Automatic Control III Lecture 6 Linearization and phase portraits. Summary of lecture 5 Thomas Schön Division of Systems and Control Department of Information Technology Uppsala University.. General properties 3. Linearization and stationary points. Phase portraits Email: thomas.schon@it.uu.se, www: user.it.uu.se/~thosc Summary of lecture 5 (I/III) 3(7) Summary of lecture 5 (II/III) (7) H, H synthesis pros and cons: H and H synthesis: Make W u G wu, W S S, W T T small. H : Minimize ( W u G wu + W SS + W TT ) dω. H : Set an upper bound for W u G wu, W S S, W T T for all ω. Results in algebraic Riccati equations. (+) Directly handles the specifications on S, T and G wu (+) Let us know when certain specifications are impossible to achieve (via γ). (+) Easy to handle several different specifications (in the frequency domain) (-) Can be hard to control the behaviour in the time domain in detail. (-) Often results in complex controllers (number of states in the controller = number of states in G, W u, W S, W T ).
Summary of lecture 5 (III/III) 5(7) DC motor with a saturated control signal (I/V) 6(7) r ũ u y Σ K(s + ) s + s(s + ) Linear multivariable controller synthesis summary:. Perform an RGA analysis. Use simple SISO controllers of PID type if the RGA analysis indicates that it might be possible. 3. Otherwise make use of LQ, MPC or H /H synthesis. DC motor controlled using a lead controller. We want to control the motor angle. The saturation ũ ũ, u = sat(ũ) = ũ >, ũ <, renders the system nonlinear. DC motor with a saturated control signal (II/V) 7(7) DC motor with a saturated control signal (III/V) 8(7) Step responses for two different amplitudes of the reference r signal. Blue: Amplitude Red: Amplitude 5 (scaled with /5) Conclusion: The step response is amplitude dependent. If the system would have been linear the two step responses would have coincided....8.6.. 6 8 Red: Reference signal r. Blue: Output signal y. 9 8 7 6 5 3 6 8. 6 8 Both the ramp and the sine responses are roughly the same as for a linear system..6.5..3...
DC motor with a saturated control signal (IV/V) 9(7) Red: r. Blue: y. Green: y when r is a ramp (same as on the previous slide). DC motor with a saturated control signal (V/V) (7) Red: before the saturation (ũ). Blue: after the saturation (u). 8 3 6 6 8 Something happens here: There is no sine present in the response and the ramp error has increased... This violates the superposition principle and the frequency fidelity! 6 8 Linearization of a nonlinear system (7) We can approximate a nonlinear system ẋ = f (x, u), y = h(x, u), by linearizing the system around an equilibrium (stationary) point (x, u ). Intuitively this amounts to approximating the system by a flat hyperplane (straight line in the scalar case). Let x(t) = x(t) x, u(t) = u(t) u, y(t) = y(t) y. A Taylor expansion (only keeping the linear terms) results in A B {}}{{}}{ d dt x = f (x, u ) x y = h(x, u ) x } {{ } C x + f (x, u ) u x + f (x, u ) u } {{ } D u, u. Phase portraits for linear systems (7) Sign: Is the solution moving towards the origin or away from the origin (along the eigenvector)? Relative size: fast and slow eigenvectors, which is dominating the solution behaviour for t and t? Complex/real: Complex conjugated eigenvalues results in circles and spirals. Cases to consider:. Two distinct real valued eigenvalues implies two eigenvectors.. One multiple eigenvalue implies one eigenvector. 3. Complex eigenvalues.
Two distinct eigenvalues with the same sign 3(7) The solution is x(t) = c e λ t v + c e λ t v. Stable node: For eigenvalues λ < λ <. The first term dominates for small t, the second term dominates for large t. Unstable node: For eigenvalues < λ < λ. Also here, the first term dominates for small t, the second term dominates for large t. Two distinct eigenvalues with opposite sign (7) For eigenvalues λ < < λ (with corresponding eigenvectors v, v ) the solution is x(t) = c e λ t v + c e λ t v. Trajectories close to v will approach the origin. v is called the stable eigenvector. Trajectories close to v will move away from the origin. v is called the unstable eigenvector. A multiple eigenvalue 5(7) For multiple eigenvalues λ = λ. Two examples in 3D 6(7) Example of a generalization to 3D..8.8.6.6.....5.5.5.5.5...6.8 Stable node (unstable: change direction). Stable star node (unstable: change direction). Left: Focus + one real eigenvalue. Right: Three real eigenvalues.
A few concepts to summarize lecture 6 7(7) Equilibrium points: An equilibrium point is a point x, u where the system is in rest, i.e. f (x, u ) =. Also referred to as stationary points. Linearization: Find a Taylor expansion of the nonlinear system around an equilibrium point and only keep the linear parts. This means that we are approximating the system using a flat hyperplane. Phase plane: A two dimensional state space that is simple to visualize graphically. Phase portraits: A plot where one state variable is plotted against another state variable. Limit cycle: A limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches ± infinity.