ELEC4631 Chap.12 Reference Tracking and Nonlinear Lyapunov Feedback
Reference Tracking and Nonlinear Lyapunov Feedback
A stabilising regulator drives deviations toward zero, but a nonzero reference requires a compatible equilibrium. For constant r, solve 0=Ax_ss+Bu_ss and r=Cx_ss+Du_ss before choosing feedforward. A static prefilter N in u=-Kx+Nr can give nominal zero steady-state error when the relevant closed-loop DC gain is nonzero; a general equilibrium-feedforward form u=u_ss-K(x-x_ss) makes dimensions and operating point explicit. Static feedforward does not guarantee rejection of constant disturbance or model mismatch—integral action supplies the internal model for that stronger goal. The nonlinear half returns to a controlled vector field x-dot=f(x)+g(x)u. Choose a positive V, expose how u enters V-dot, then cancel or dominate terms and inject damping. Substitute the controller into the original model and state local/global, nominal/robust and regularity boundaries.
What this chapter covers
- 01Regulation versus nonzero tracking
- 02Steady-state plant/output equations
- 03SISO static reference prefilter
- 04Equilibrium feedforward and deviation coordinates
- 05Estimated-state tracking and separation
- 06Control-affine nonlinear systems
- 07Lyapunov derivative shaping, cancellation and damping
Nominal constant-reference prefilter
- closureSubstitute the control law: x-dot=-5x+Nr.
- equilibriumAt constant steady state, 0=-5x_ss+Nr, so x_ss=(N/5)r.
- prefilterBecause y_ss=x_ss, the condition y_ss=r requires N=5.
- stabilityWith N=5 and constant r, define deviation z=x-r. Then z-dot=-5z, so tracking error decays exponentially.
- scopeThe conclusion is nominal constant-reference tracking. A plant gain error or constant disturbance can reintroduce offset without integral action.
Key terms
- Regulation
- Driving the state or output deviation toward zero, usually with a stabilising feedback law.
- Reference tracking
- Making the output approach a commanded nonzero signal while preserving stable error dynamics.
- Compatible equilibrium
- A pair (x_ss,u_ss) satisfying both the steady plant equation and desired output equation for the reference.
- Prefilter
- A feedforward factor multiplying r so the nominal stabilised loop has the desired constant steady-state gain.
- Integral action
- An augmented state accumulating tracking error, providing an internal model for constant references/disturbances under suitable closed-loop design.
- Control-affine system
- A nonlinear model x-dot=f(x)+g(x)u in which the input enters linearly even though state dependence may be nonlinear.
- Derivative shaping
- Choosing u(x) so the Lyapunov derivative becomes negative in the intended domain, often by cancellation plus damping.
Reference Tracking and Nonlinear Lyapunov Feedback FAQ
Why does stabilising K not automatically track r?
With u=-Kx and no feedforward, the stable closed-loop equilibrium is normally x=0 and hence y=0. A nonzero command must be embedded through a compatible steady state, prefilter, integral state or another reference model.
When does the SISO prefilter formula fail?
It requires a stable nonsingular A-BK and a nonzero closed-loop DC path from the feedforward input to output. If the reference is not achievable or D/dimensions differ, solve the stacked equilibrium equations instead of dividing by a memorised scalar.
Does a prefilter reject a constant disturbance?
Not generally. It corrects the nominal reference gain. Model mismatch or an added constant disturbance changes the equilibrium and can create offset. Integral action introduces the internal model needed for robust constant-signal rejection under appropriate assumptions.
Is nonlinear cancellation robust?
Exact cancellation is a nominal statement unless uncertainty is modelled. After choosing u(x), substitute it into the real vector field, compute V-dot and state the domain. Saturation, parameter error or a vanishing input channel can weaken the certificate.
What is a complete linear tracking solution?
Write the stabilised law and solve the steady plant/output equations for the desired constant r. If using a scalar prefilter, derive N from the actual closed-loop DC map rather than quoting a formula whose D or sign conventions may differ. Check the relevant inverse and nonzero gain conditions. Define deviation z=x-x_ss and show z-dot=(A-BK)z, then use the already verified Hurwitz property to conclude nominal convergence. With an observer, ensure plant and observer receive the same feedforward input and use separation for the augmented deviation/error dynamics. Finish by stating that a static prefilter corrects nominal reference gain but does not automatically reject disturbance or parameter error; integral action is the natural stronger structure for constant offset rejection.
What is a complete nonlinear feedback solution?
Find the controlled-system equilibrium and choose a positive candidate V on a declared domain. Compute V-dot=grad(V)^T[f(x)+g(x)u] without inserting a controller prematurely, so the control channel is visible. Choose u(x) to cancel known terms, dominate cross terms or inject damping. Substitute the proposed law into the original vector field and recompute V-dot completely. Check negative definiteness or use LaSalle for a semidefinite result, then state local/global and nominal/robust scope. Discuss whether g(x) vanishes, whether the law is continuous and whether actuator saturation invalidates the algebra. The controller value alone is not the design result; the resulting derivative and theorem conditions are.
What paired practice connects the two halves of this chapter?
For a scalar unstable linear plant, first choose K for regulation and derive N for a constant reference. Add a constant disturbance and show the steady offset, then augment an integral state and discuss the new design permission. Next replace the plant with x-dot=x^3+u, choose V=x^2/2 and design cancellation plus damping. In both problems identify the target equilibrium, derive error or energy dynamics and state nominal scope. Ask an AI tutor to perturb the plant coefficient after your design and predict which equilibrium or derivative term no longer cancels. This comparison reveals the shared structure: specify a target, stabilise deviation and verify the original model.
Exam move
For every tracking problem, write the two steady-state equations before the control law. Decide whether the requested output is achievable, then define deviation coordinates and recover A-BK as the error matrix. Practise both a scalar prefilter and a multi-equation equilibrium-feedforward setup. Add an observer only after the target equilibrium and estimator input are consistent, then use separation for nominal error dynamics. For nonlinear synthesis, start with a positive radially meaningful V, write V-dot with the input term exposed and choose cancellation/damping transparently. Verify equilibrium, derivative sign, domain, regularity and actuator assumptions. Finish every solution by stating whether the result is nominal or robust and local or global.
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