Topics in Nonlinear Systems

18 March 2023 - 25 mins read time
Tags: control theory nonlinear systems Lyapunov stability feedback linearization backstepping sliding mode ADRC fuzzy control

This note is adapted from course materials for EL2620 Nonlinear Control at KTH Royal Institute of Technology. Instructor: course teaching staff.

1. Why Nonlinear Systems Matter

Linear models are powerful because they are analyzable and often accurate near an operating point. But physical systems are nonlinear by default.

Nonlinearity enters through:

That is why nonlinear control is not a niche afterthought. It is the broader setting in which linear control is a local approximation.

2. Nonlinear State Models, Equilibria, and Linearization

A general nonlinear control system is written as

\[\dot x=f(x,u,t), \qquad y=h(x,u,t).\]

For a constant input $u_e$, an equilibrium $x_e$ satisfies

\[f(x_e,u_e,t)=0.\]

Different equilibria can have different stability properties, so the operating point matters.

2.1 Jacobian linearization

Define perturbations

\[\delta x=x-x_e, \qquad \delta u=u-u_e, \qquad \delta y=y-y_e.\]

Then near the equilibrium,

\[\delta \dot x \approx A \delta x + B \delta u, \qquad \delta y \approx C \delta x + D \delta u,\]

with Jacobians

\[A=\left.\frac{\partial f}{\partial x}\right|_{(x_e,u_e)}, \quad B=\left.\frac{\partial f}{\partial u}\right|_{(x_e,u_e)},\] \[C=\left.\frac{\partial h}{\partial x}\right|_{(x_e,u_e)}, \quad D=\left.\frac{\partial h}{\partial u}\right|_{(x_e,u_e)}.\]

This is the bridge from nonlinear physics to linear design. It is powerful, but only local.

2.2 What linearization can and cannot say

Linearization can reveal:

It cannot guarantee:

3. Phase Plane and Geometric Intuition

For second-order systems, phase-plane plots turn dynamics into geometry:

This viewpoint is helpful because it makes nonlinear motion visual:

Phase-plane intuition is one of the cleanest ways to feel the difference between local linear reasoning and global nonlinear behavior.

4. Lyapunov Stability: Direct Method, Invariance, and Barbalat

For nonlinear systems, poles are not enough. Stability must be defined directly in state space.

4.1 Core definitions

Common notions are:

4.2 Lyapunov direct method

Take a scalar function $V(x)$ that behaves like an energy:

\[V(x)>0 \text{ for } x \neq 0, \qquad V(0)=0.\]

If

\[\dot V(x)=\nabla V^\top f(x) \le 0,\]

then trajectories cannot move uphill in $V$.

4.3 A standard lemma

Lemma. Suppose $V \in C^1$ satisfies

\[\alpha_1(\|x\|) \le V(x) \le \alpha_2(\|x\|)\]

for class-$\mathcal{K}_\infty$ functions $\alpha_1,\alpha_2$, and

\[\dot V(x) \le -\alpha_3(\|x\|)\]

for another class-$\mathcal{K}_\infty$ function $\alpha_3$. Then the origin is globally asymptotically stable.

Proof sketch. Positive definiteness and properness imply bounded level sets, so trajectories remain bounded. Since $V$ decreases strictly away from the origin, the state must move toward the largest invariant set inside ${\dot V=0}$, which here is only the origin. That gives global attractivity and stability.

This is the main pattern behind many nonlinear proofs: invent an energy-like quantity and show it decreases.

4.4 LaSalle and Barbalat

Two recurring tools are:

LaSalle helps when $\dot V \le 0$ but not strictly negative. Barbalat helps when we know a signal is bounded, integrable, and sufficiently regular, so it must converge to zero.

These are the workhorses behind many nonlinear and adaptive stability arguments.

4.5 What Stability Theory Does and Does Not Give You

Three frameworks organize sufficient conditions, from coarse to fine:

Framework Measures Strength Weakness
Small gain norms ($\mathcal L_2$ gain, $H_\infty$) simple, robust-control friendly conservative: loop gain below 1 at every frequency
Passivity input-output energy (inner product) uses phase as well as gain sits between the others; fewer off-the-shelf results
Lyapunov energy of every state most general and structural needs a function valid at all states

All three are sufficient, and the order small gain $\succ$ passivity $\succ$ Lyapunov is the typical order of decreasing conservatism (not a theorem for every system). Stability words are not interchangeable:

Notion What it says What it does not say
Local (Lyapunov) stability trajectories starting close to an equilibrium stay close nothing about distant initial conditions or convergence
Asymptotic stability local stability plus convergence to the equilibrium the basin may be small; convergence may be slow
Global asymptotic stability convergence from every initial condition not exponential; not robust to disturbances without more
Exponential stability convergence at a rate bounded by an exponential holds only on the stated region unless global
Boundedness / uniform ultimate boundedness states stay in a bounded set, or enter a ball no convergence to a point
Input-to-state stability (ISS) bounded input gives bounded state, with $x\to0$ when input $\to0$ does not give robustness to unmodelled dynamics or parameter change
Input-output ($\mathcal L_p$) stability bounded input energy gives bounded output energy says nothing about internal states unless detectable and stabilizable
Robust stability stability for every model in a stated uncertainty set only for that set; not a statement about the nominal model alone

Each follows from the previous only with extra assumptions (for example, local asymptotic stability does not give global stability; asymptotic stability of the nominal model does not give ISS or robust stability). Further caveats:

4.6 How the Nonlinear Toolbox Is Organized

A nonlinear-systems course usually has four parts of very different practical value, and knowing which is which saves time:

  1. Catalogue of nonlinear phenomena and their models (saturation, dead zone, friction, backlash, hysteresis, limit cycles, bifurcations, chaos). This is the part most directly useful on hardware.
  2. Stability theory (Lyapunov, passivity, small gain). It decides when a conclusion is correct, much as the if-part of a theorem in analysis does, but it rarely produces a design number.
  3. Geometric theory on manifolds (differential-geometric controllability and observability, normal forms). It is mathematically deep and rarely used directly in practice.
  4. Synthesis methods (feedback linearization, backstepping, sliding mode). They exist, but each rests on assumptions (exact model, matching, full state) that are hard to meet.

Dimension is the main source of difficulty, not chaos alone. A three-state system such as the Lorenz equations can be chaotic and still be well studied, and a system with supercritical Hopf bifurcation produces gentle oscillations that are easy to analyze and sometimes to control. The planar and low-order cases are the ones with complete theory. For the general three-body problem (18 states), no closed-form solution exists, and analysis has moved from solving to statistics, numerics, and qualitative structure.

The Lorenz attractor Figure 1. The Lorenz attractor, a three-state chaotic system $\dot x=\sigma(y-x)$, $\dot y=x(\rho-z)-y$, $\dot z=xy-\beta z$. Trajectories never repeat, yet stay on a bounded set with a well-understood geometry. Source: File:Lorenz attractor yb.svg. License: CC BY-SA 3.0.

Another useful distinction is between analysis methods that ignore the problem structure and those that exploit it. The describing function, for example, is an approximate frequency-domain method with no existence guarantee, while Lyapunov and invariance arguments are rigorous but may be conservative. Choose the method for the question: use the approximate one to guess a limit-cycle amplitude, then use simulation and hardware to confirm.

5. Limit Cycles, Friction, and Strong Nonlinearities

5.1 Where Real Jitter Comes From: Limit Cycles

At rest or low speed, small persistent vibration (steering wheel at standstill, a lane-keeping loop going straight, a servo parked at its target) is often a limit cycle rather than sensor noise passed through a linear loop, but this is a hypothesis to test, not a classification. The two checks below give diagnostic evidence only; each can mislead (nonlinear loops can also amplify noise, and a linear loop can resonate with a disturbance near its bandwidth), so confirm with several experiments.

  1. Frequency test. A linear time-invariant loop does not create new frequencies, so a disturbance or noise at frequency $\omega$ appears at $\omega$ at the output. If the observed shake is at a frequency that the sensor noise and disturbances do not contain, a nonlinear mechanism is suspected. Because the noise spectrum is rarely measured exactly, treat a mismatch as suggestive.
  2. Gain sweep. Raising loop gain at the offending frequency should shrink the effect of a linear disturbance. Limit cycles typically change amplitude or frequency with gain and often disappear when the loop is opened or the gain is cut. Several mechanisms can show the same trend, so a sweep narrows the candidates without identifying the cause.

Friction is the most common source. A standard static friction model combines Coulomb friction, a Stribeck (velocity-weakening) region, and viscous friction:

\[F(v)=\Big[F_c+(F_s-F_c)\,e^{-(v/v_s)^2}\Big]\operatorname{sgn}(v)+\sigma\,v,\]
with Coulomb level $F_c$, static (breakaway) level $F_s>F_c$, Stribeck velocity $v_s$, and viscous coefficient $\sigma$. The slope $dF/dv$ is negative for $v$ just above zero. Linearized about a sliding speed $v_0$, friction acts as a damper of coefficient $\sigma_{eff}=dF/dv {v_0}$, so where the Stribeck term dominates, $\sigma{eff}<0$: the mechanism injects energy into small speed fluctuations instead of dissipating it, and a velocity or position loop with too little gain in that band can sustain a stick-slip oscillation. That is why more gain in the sticking speed band can reduce the oscillation, while more gain at standstill with dead zone or Coulomb friction can increase it.

Stribeck curve: friction versus speed Figure 2. Stribeck curve: friction against sliding speed. Region 1 is static friction, region 2 is the velocity-weakening (negative-slope) region responsible for stick-slip, and region 3 is viscous-dominated friction. Source: File:Curva di Stribeck.svg. License: CC BY-SA 3.0.

Typical sources and their signatures:

Source Typical symptom Gain dependence
Dead zone (torque sensor, bridge dead time, assist curve, current sampling) small oscillation about the target at rest grows with gain
Coulomb friction with integral action hunting around setpoint grows with integral gain
Stribeck (velocity-weakening) friction stick-slip at low speed worsens at low gain, reduced by more gain in the sticking band
Backlash/gap chaotic or periodic impacts at particular speeds, with bifurcations as speed or gain varies depends on operating speed
Variable gain through rising and falling segments small oscillation before instability grows as gain rises

The describing-function (harmonic-balance) view gives a useful prediction: a limit cycle of amplitude $A$ and frequency $\omega$ is suggested where $1+N(A)G(j\omega)=0$. This is an approximate condition that keeps only the first harmonic and assumes the linear part filters higher harmonics; it is neither a rigorous existence proof nor a stability test for the cycle, and it can give false positives and false negatives. If a gain map or friction-compensation curve is non-monotonic, $N(A)$ can sweep through the critical point, which makes a cycle plausible; confirm with simulation of the nonlinear model or hardware tests. Design gain tables and stacked “add-this-subtract-that” gain modifications so that the resulting curve is monotonic, then verify on hardware.

Practical rule on switches and mode logic: every extra switch is a potential chatter source. Add it only after analyzing the closed loop in each mode and across the switch (a common Lyapunov function or a dwell-time argument), and prefer smooth schedules.

Describing-function analysis in detail. Consider the loop of a static nonlinearity $\mathrm{NL}$ in series with a linear block $G_0(s)$ under unity negative feedback:

Nonlinear element in series with a linear block in a feedback loop Figure 3. The describing-function setup: a static nonlinearity NL followed by a linear block $G_0(s)$ in a feedback loop. Source: File:Describing function method.png. License: CC0.

Drive the nonlinearity with $e(t)=A\sin\omega t$. The output $u(t)$ is periodic, with Fourier series $u(t)=\sum_k\big(a_k\sin k\omega t+b_k\cos k\omega t\big)$. The describing function keeps only the fundamental:

\[N(A)=\frac{a_1+jb_1}{A},\qquad a_1=\frac1\pi\int_0^{2\pi}u(A\sin\theta)\sin\theta\,d\theta,\quad b_1=\frac1\pi\int_0^{2\pi}u(A\sin\theta)\cos\theta\,d\theta .\]

If $G_0(j\omega)$ attenuates harmonics (a low-pass assumption), a sustained oscillation requires the loop to return the fundamental unchanged:

\[1+N(A)\,G_0(j\omega)=0\quad\Longleftrightarrow\quad G_0(j\omega)=-\frac{1}{N(A)} .\]

Plot the locus $-1/N(A)$ against the Nyquist curve of $G_0$; an intersection gives a predicted amplitude $A$ and frequency $\omega$. Standard describing functions for common elements (with $N$ real for odd, memoryless nonlinearities):

Element Definition $N(A)$
Relay (ideal, level $M$) $u=M\,\mathrm{sgn}(e)$ $\dfrac{4M}{\pi A}$
Saturation (slope $k$, linear range $a$) $u=k\,\mathrm{sat}_a(e)$ $\dfrac{2k}{\pi}\Big[\arcsin\dfrac aA+\dfrac aA\sqrt{1-\dfrac{a^2}{A^2}}\Big]$ for $A\ge a$
Dead zone (slope $k$, half-width $d$) $u=k(e-d\,\mathrm{sgn}\,e)$ for $\lvert e\rvert>d$ $k-\dfrac{2k}{\pi}\Big[\arcsin\dfrac dA+\dfrac dA\sqrt{1-\dfrac{d^2}{A^2}}\Big]$ for $A\ge d$

I checked the three expressions by numerical Fourier integration: for $k=a=d=1$ and $A=1.5$ they give $N=0.781$ (saturation) and $0.219$ (dead zone), and $N=0.637$ for a unit relay at $A=2$, all matching the closed forms to 9 digits. Saturation and dead zone satisfy $N_{sat}+N_{dz}=k$, as expected because their outputs add to the input.

Worked example: relay feedback. Take $G_0(s)=\dfrac{1}{s(s+1)^2}$ with a unit relay ($M=1$). The phase of $G_0(j\omega)$ is $-90^\circ-2\arctan\omega$, which reaches $-180^\circ$ at $\arctan\omega=45^\circ$, i.e. $\omega=1$ rad/s, where $ G_0(j1) =\dfrac{1}{1\cdot2}=0.5$. The condition $G_0(j\omega)=-1/N(A)$ requires $-0.5=-\dfrac{\pi A}{4M}$, so
\[\omega_{pred}=1\ \text{rad/s},\qquad A_{pred}=\frac{2M}{\pi}=0.637 .\]

Direct time simulation of the relay loop (unity negative feedback, step $0.01$ s, after the transient) gives an output oscillation with amplitude $0.658$ and period $6.42$ s, against predicted $0.637$ and $2\pi=6.28$ s. The prediction is within about 3% in amplitude and 2% in period: this is the typical accuracy of the method, good enough to size an oscillation and poor as a proof. The residual is the higher harmonics the method discards, and when the linear part does not filter them well the error can be large, which is why the method should be confirmed by simulation or test.

A limit cycle of a relaxation (van der Pol type) oscillator Figure 4. Phase portrait of a slow-fast (van der Pol type) limit cycle: a closed, isolated trajectory that attracts nearby trajectories. This is the geometric object that describing-function analysis tries to locate. Source: File:Van der Pol type slow-fast limit cycle.svg. License: CC0.

5.2 Trade-Offs for Strong Nonlinearities

When friction, dead zone, or backlash is strong, no controller delivers both precision and quiet; the realistic move is to choose which to give up, in a specific place:

Problem Measure Cost
Coulomb-friction hunting at setpoint set gain to zero inside a small error band larger steady-state error
Stick-slip in the speed loop raise gain in the sticking speed range, restore outside it more noise
Dead zone freeze or switch off integrator inside it slower response
Backlash scale gain when in the gap (if measurable); otherwise avoid the limit-cycle region tracking not guaranteed

If precision and quietness both fail with correct engineering, the plant needs improvement (lower friction, tighter backlash, smaller dead zone).

The weak/strong distinction also matters for identification, but strong nonlinearities are not inherently unidentifiable. Whether a gap size or a static-friction level can be recovered depends on the assumed model structure (for example a Stribeck or LuGre friction model, or a backlash model), on whether the parameters are structurally identifiable, on excitation that actually exercises the nonlinearity (velocity reversals, small and large amplitudes), on which signals are measured (position, velocity, torque), and on the experimental conditions. Dedicated experiments such as slow ramps or velocity-reversal tests can identify friction and backlash parameters in many setups. What does fail is fitting a linear model to a sine sweep of a nonlinear system: the coefficients then describe a describing-function-like average at one operating amplitude and have no direct physical meaning. Choose the model structure first, excite the nonlinearity deliberately, and validate on data not used in fitting.

6. Gain Scheduling and Hybrid Linear-Nonlinear Design

Many real plants operate over wide envelopes where one linear controller is not enough. Gain scheduling uses a family of local linear controllers indexed by operating condition:

This is not a fully nonlinear synthesis method, but it is often the most practical compromise between fidelity and complexity.

The design risks are:

This is why gain scheduling is often paired with a hybrid linear-nonlinear analysis mindset rather than treated as a pure plug-and-play recipe.

7. Feedback Linearization and Dynamic Inversion

If the nonlinear structure is known accurately enough, we can try to cancel it explicitly.

7.1 Basic idea

Suppose

\[\dot x = f(x)+g(x)u.\]

If the input enters in a suitable way, choose

\[u=\alpha(x)+\beta(x)v\]

so the transformed dynamics become approximately linear in the new input $v$.

This is the core of feedback linearization and dynamic inversion.

7.2 Why it is powerful

If it works well:

7.3 Why it is fragile

If the model is wrong:

So dynamic inversion is powerful when the model is reliable and the bandwidth is realistic. It is dangerous when it pretends uncertainty is negligible.

8. Backstepping

Backstepping is one of the most systematic nonlinear-design methods for strict-feedback systems.

Consider the two-state template

\[\dot x_1 = f_1(x_1) + g_1(x_1)x_2,\] \[\dot x_2 = f_2(x) + g_2(x)u.\]

The idea is recursive:

  1. treat $x_2$ as a virtual control for the $x_1$-subsystem,
  2. design a stabilizing virtual control $\alpha_1(x_1)$,
  3. define the error
\[z_2 = x_2 - \alpha_1(x_1),\]
  1. design the real input $u$ so both $z_1$ and $z_2$ decay.

8.1 A representative theorem

Theorem. For a strict-feedback system with smooth known nonlinearities and nonvanishing input gains, a recursive backstepping design can produce a control law and Lyapunov function such that the origin is asymptotically stable.

Proof sketch. Start with a Lyapunov candidate $V_1(z_1)$ for the first subsystem. Introduce a virtual control to make $\dot V_1$ negative up to a residual involving $z_2$. Then augment the Lyapunov function to

\[V_2 = V_1 + \frac{1}{2} z_2^2\]

and choose $u$ so the cross terms cancel and $\dot V_2$ becomes negative definite. The same pattern extends recursively.

Backstepping is attractive because it builds the controller and the proof together.

8.2 Engineering reading of backstepping

Backstepping is especially useful when:

This is why it reappears in adaptive control, underactuated systems, and nonlinear robotics.

9. Sliding Mode Control

Sliding mode control is a variable-structure method that drives the system onto a chosen sliding surface and then keeps it there.

Choose a surface such as

\[s(x)=0.\]

A typical reaching law aims for

\[\dot V = \frac{d}{dt}\left(\frac{1}{2}s^2\right)=s \dot s \le -\eta |s|\]

for some $\eta>0$.

This yields two phases:

Strengths:

Weaknesses:

Practical designs often smooth the sign function or use higher-order sliding modes to reduce chattering.

Equations of the ideal design. For a plant $\dot x=f(x)+b(x)u+d$ with matched bounded disturbance $ d \le D$, choose a sliding variable such as $s=\dot e+\lambda e$ for a tracking error $e$. Choose the control
\[u=u_{eq}-k\,\mathrm{sgn}(s),\qquad k>D+\eta,\]
where $u_{eq}$ solves $\dot s=0$ for the nominal plant (the continuous control that would keep the state on the surface). Then $s\dot s\le-\eta s $, so $s$ reaches zero in finite time $t_r\le s(0) /\eta$, and on the surface $e$ decays as $e^{-\lambda t}$ regardless of $d$.

Reaching and sliding phases of first-order sliding mode control Figure 5. Phase plane of first-order sliding-mode control: trajectories first move toward the sliding surface $s=0$ (reaching phase), then slide along it toward the origin (sliding phase), with visible chatter along the line. Source: File:First order sliding mode control.svg. License: CC BY-SA 3.0.

Sampled implementation, in numbers. The simplest case $\dot x=u+d$, $u=-k\,\mathrm{sgn}(x)$ with sampling period $T$ becomes $x_{j+1}=x_j+T\big(-k\,\mathrm{sgn}(x_j)+d_j\big)$. With $d=0$, $k=2$, $T=0.05$ s the state does not settle to zero: it alternates between two values $kT=0.1$ apart, a peak-to-peak chatter of exactly $kT$. With a bounded disturbance $ d \le0.5$ the simulated state stays within $\pm(k+D)T=\pm0.125$ (simulated maximum $0.122$). The chatter amplitude is proportional to the switching gain and the delay or sampling time, so reducing it means reducing $k$ toward the true $D$ (which needs a good disturbance bound), shortening $T$, or replacing $\mathrm{sgn}$ by a boundary layer. In the ideal theory these quantities are zero or infinite; in a real loop they are the design variables.

Why sliding mode is rarely the default in industry. In more detail:

Use sliding mode where matched uncertainty is large, the actuator is fast and tolerant (power electronics, motor currents), and the sliding variable is measured or can be estimated within the design’s assumptions.

10. Bang-Bang Control

Bang-bang control uses extreme actuator values:

\[u(t) \in \{u_{\min}, u_{\max}\}.\]

This appears naturally when:

Bang-bang laws sit at the boundary between nonlinear control and optimal control:

For the optimal-control version, see Adaptive, Optimal, Robust, and Learning Control.

11. Fuzzy Control

Fuzzy control is attractive when the system is hard to model precisely but expert rules are available.

The logic is:

Strengths:

Weaknesses:

Fuzzy control can work very well in industry, but it is not a substitute for structural stability analysis when guarantees matter.

12. Active Disturbance Rejection Control

ADRC treats unknown plant mismatch and disturbances as an aggregated disturbance to be estimated and cancelled online.

The usual ingredients are:

12.1 Why ADRC is appealing

12.2 What to be careful about

Conceptually, ADRC sits between model-based feedback linearization and robust disturbance-observer design.

13. Nonlinear Controllability and Observability

The linear rank tests do not fully settle nonlinear controllability or observability.

At the nonlinear level, useful ideas include:

For many engineering purposes, linearization still gives the first useful answer. But nonlinear systems force more precise language: “controllable near which point, in what sense, and under what input constraints?”

14. Reading Map Across the Control Series

This note is the nonlinear-design bridge in the control series:

15. Compact Recall Map

The shortest useful nonlinear-control workflow is:

  1. write the nonlinear model and identify the operating point,
  2. separate local questions from global ones,
  3. use Jacobian linearization when a local controller is enough,
  4. use Lyapunov geometry when guarantees matter,
  5. choose a nonlinear design method that matches the structure:
    • gain scheduling for wide but mostly linear regimes,
    • feedback linearization for accurately known nonlinearities,
    • backstepping for cascaded strict-feedback systems,
    • sliding mode for matched uncertainty,
    • ADRC when disturbance estimation is central.

Nonlinear systems are not exceptions to control theory. They are the full problem. Linear theory is the local window that makes pieces of the full problem manageable.