Nonlinear Control Systems

Twenty-eight chapters have been built on one property. Chapter 3 defined the transfer function by assuming that the response to a sum of inputs is the sum of the responses; Chapter 14 defined the frequency response by assuming that a sinusoid in produces a sinusoid out at the same frequency; the root locus, the Nyquist criterion and the state transition matrix all inherit that assumption. Superposition is not one tool among many — it is the foundation the entire edifice stands on.

No physical system obeys it. An amplifier clips. A valve cannot open beyond its stop. A gear train has lash. Static friction resists motion until it suddenly does not. These are not modelling refinements to be added later; they are the reason a controller that looks perfect on paper hunts, sticks or oscillates on the bench. When superposition fails, four behaviours become possible that no linear model can produce.

The limit cycle deserves particular attention, because it is the phenomenon this chapter spends most of its effort on. In a linear system a sustained oscillation requires a pole pair sitting exactly on the imaginary axis — an infinitely improbable condition — and even then the amplitude is whatever the initial condition made it. A limit cycle is different in kind: it is an isolated closed trajectory that nearby motions converge to. Its amplitude and frequency are properties of the system, not of how it was started. That is why a relay-controlled thermostat always hunts with the same swing, and why a servo with backlash always buzzes at the same frequency however it is disturbed.

In practice a small catalogue covers most of what is met. Each is described by a static input–output characteristic \(y=f(x)\), and the single most useful thing to notice about any of them is whether that characteristic is single-valued or multi-valued. A single-valued characteristic — saturation, dead zone, an ideal relay — assigns one output to each input, and the corresponding describing function of Section 29-5 turns out to be a real number: pure amplitude-dependent gain, no phase shift. A multi-valued one — backlash, a relay with hysteresis — depends on the direction of travel, which is to say it has memory, and its describing function is complex: it contributes phase lag, and phase lag is what destabilises loops.

Almost all of these can be pushed into a single standard arrangement: a static nonlinearity \(f(\cdot)\) in the forward path, with everything else — plant, sensor, linear compensator — lumped into one linear transfer function \(G(s)\) around it. That configuration is called the Lur'e problem, and every analytical method in the rest of this chapter except the phase plane is built for it.

The first move is always the cheapest one: pretend the nonlinearity is not there, and see how far that carries. Chapter 2 did exactly this for physical models, expanding each nonlinear relation about an operating point and keeping the first term; Chapter 23 rewrote the procedure in matrix form. It is worth restating precisely, because knowing what it proves matters more than knowing how to do it.

Given \(\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})\) with an equilibrium \(\mathbf{x}^{*}\) satisfying \(\mathbf{f}(\mathbf{x}^{*})=\mathbf{0}\), write \(\mathbf{x}=\mathbf{x}^{*}+\delta\mathbf{x}\) and expand:

Three limitations follow, and each one motivates a later section. First, the result is local: a system can be beautifully stable for small disturbances and diverge for larger ones, which is precisely the amplitude-dependence of Section 29-1. Estimating the region of attraction needs Lyapunov's direct method, Section 29-7. Second, the borderline case genuinely occurs — Example 5 gives a system whose linearisation is a pure centre, undecidable by eigenvalues, yet which is globally asymptotically stable. Third, and most importantly for control engineering, the hard nonlinearities of Section 29-2 are not differentiable at the origin at all. A relay has no Jacobian; neither does backlash. For those the linearised model does not merely mislead, it does not exist. That is what the phase plane and the describing function are for.

For a second-order system there is an exact method that needs no linearity at all: draw the motion. Take the state variables \(x_1=x\) and \(x_2=\dot x\) of Chapter 23 and plot \(x_2\) against \(x_1\) as time runs. The resulting curve is a trajectory, and the family of trajectories from all initial conditions is the phase portrait. Everything about the system's behaviour is visible in it.

Time is eliminated by dividing one state equation by the other. For \(\ddot x=f(x,\dot x)\),

Two consequences are immediate and worth having at your fingertips. Where \(x_2>0\) the trajectory moves to the right, and where \(x_2<0\) it moves to the left — the upper half of the plane is always traversed rightward. And wherever a trajectory crosses the \(x_1\) axis with \(f\ne0\), the slope is infinite, so it crosses perpendicularly. Sketching a portrait usually starts from those two facts, refined by the isocline method: setting the slope to a constant \(m\) gives the curve \(f(x_1,x_2)=mx_2\), along which every trajectory has slope \(m\); drawing a handful of isoclines with short slope marks lets the trajectories be threaded in by eye.

The points where both derivatives vanish are the singular points — the equilibria. Near each of them the linearisation of Section 29-3 applies, and its eigenvalues fix the local shape. There are six possibilities, and recognising them from a sketch is a standard examination question.

The saddle point is the one with no linear-systems intuition behind it, because a linear system has only one equilibrium and a saddle needs neighbours. Its two incoming directions form a curve — the separatrix — that divides the plane into regions whose trajectories end up in entirely different places. When a nonlinear system has several equilibria, it is the saddles between them that carve the state space into basins of attraction.

A limit cycle appears in the portrait as an isolated closed curve. If nearby trajectories spiral onto it from both sides it is stable, and the system will settle into that oscillation from any nearby start; if they spiral away it is unstable, and it marks the boundary of a region of attraction. The figure below shows the portrait of a relay-controlled inertia, where the closed curve is not isolated at all — a whole family of them fills the plane — and shows what a rate-feedback term does to it.

The frequency-response methods of Chapters 14 to 18 are the most powerful design tools in the book, and they are lost the moment a nonlinearity enters, because a sinusoid into a nonlinearity does not come out a sinusoid. The describing function recovers them by an approximation of considerable audacity: throw the harmonics away.

Drive the nonlinearity with \(x(t)=A\sin\omega t\). Its output \(y(t)\) is periodic with the same period, so it has a Fourier series \(y(t)=Y_0+Y_1\sin(\omega t+\phi_1)+Y_2\sin(2\omega t+\phi_2)+\dots\). Assume the characteristic is odd-symmetric, which kills \(Y_0\), and then simply discard every term beyond the first. What remains is a sinusoid at the input frequency, with an amplitude and a phase that depend on \(A\). Define their ratio as a gain.

The approximation stands or falls on one assumption, which must be checked in every application: the linear part of the loop must be low-pass, so that the harmonics generated by the nonlinearity are filtered out before they return to its input. Since most plants roll off at \(-40\) dB/decade or steeper, the assumption is usually reasonable — but it fails for resonant or lightly damped plants, and it fails when the nonlinearity generates a strong third harmonic that lands near a plant resonance.

Deriving a describing function is a Fourier coefficient calculation. Take the ideal relay of output \(\pm M\): the output is a square wave of amplitude \(M\) in phase with the input, whatever \(A\) is. Its fundamental coefficient is

The relay's describing function is real and falls off as \(1/A\): it presents infinite gain to a vanishingly small signal and negligible gain to a large one. Saturation takes more work but follows the same route. With slope \(K\) up to \(\pm S\) and constant beyond, and writing \(\theta_1=\sin^{-1}(S/A)\) for the angle at which the input reaches the break point,

The middle step uses \(\sin2\theta_1=2(S/A)\cos\theta_1\) to collapse the first integral. Check the limits: as \(A\to S\) the bracket tends to \(\pi/2\) and \(N\to K\), the unsaturated gain, as it must; as \(A\to\infty\) the bracket tends to zero and \(N\to0\). So saturation is a gain that falls monotonically from \(K\) to zero as the signal grows.

With \(N(A)\) in hand, the loop of Section 29-2 — nonlinearity \(f\) followed by linear \(G(s)\), closed with negative feedback — has become an ordinary linear loop whose forward gain happens to depend on the signal amplitude. Its characteristic equation is therefore the familiar one, and a sustained oscillation is exactly the condition that it be satisfied on the imaginary axis:

The equation is best solved graphically, and the picture is a direct extension of the Nyquist diagram of Chapter 15. Plot \(G(j\omega)\) as usual. On the same axes plot the locus of \(-1/N(A)\) as \(A\) runs from zero to infinity. In the linear case that locus degenerates to the single critical point \(-1\); here it is a curve, and the question "does the Nyquist plot pass through the critical point" becomes "does it cross the critical locus". Each intersection is a predicted limit cycle: \(\omega\) is read from the \(G\) curve and \(A\) from the \(-1/N\) curve.

The shape of the critical locus is characteristic of the nonlinearity. For an ideal relay \(N=4M/(\pi A)\) is real and positive, so \(-1/N=-\pi A/(4M)\) runs along the negative real axis from the origin outward — a relay loop therefore has a limit cycle whenever \(G\) crosses the negative real axis at all, which is to say almost always. For saturation \(N\) falls from \(K\) to \(0\), so \(-1/N\) runs along the negative real axis from \(-1/K\) outward, and a limit cycle exists only if \(G\) crosses the real axis to the left of \(-1/K\) — that is, only if the small-signal linear loop would have been unstable. For a relay with hysteresis the locus is a horizontal line below the real axis at \(\operatorname{Im}=-\pi h/(4M)\), as Example 4 derives.

An intersection tells you a limit cycle exists; it does not by itself tell you whether the system will settle into it. That requires a perturbation argument, and the argument is a direct reading of the Nyquist criterion of Chapter 16. Suppose the oscillation is nudged to a slightly larger amplitude. The operating point moves along the \(-1/N\) locus in the direction of increasing \(A\). If that direction takes the point into the region that \(G(j\omega)\) encircles — the region a Nyquist analysis would call unstable — the oscillation grows further and the limit cycle is not sustained at that amplitude. If it takes the point out of the encircled region, the oscillation decays back and the limit cycle is stable.

Chapter 10 defined stability through the location of poles, which presupposes a transfer function. Lyapunov's contribution, made in 1892 and largely ignored by engineers for sixty years, is a definition and a test that never mention poles, linearity, or even the solution of the differential equation.

The idea comes from mechanics. If the total energy of a system is positive whenever the system is away from rest, and if that energy is strictly decreasing along every motion, then the motion must eventually run out of energy and come to rest. Nothing in that argument requires the equations to be linear; it requires only an energy-like function. Lyapunov's insight was that any function with those two properties will do, whether or not it is a physical energy.

Let \(\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})\) with \(\mathbf{f}(\mathbf{0})=\mathbf{0}\). A scalar function \(V(\mathbf{x})\) is positive definite in a region if \(V(\mathbf{0})=0\) and \(V(\mathbf{x})>0\) for every other \(\mathbf{x}\) there. Its rate of change along the trajectories is obtained by the chain rule without solving anything:

That last sentence is the method's defining characteristic, and the source of both its power and its frustration. There is no algorithm that produces \(V\) for a general nonlinear system. What exists is a set of constructions that work for particular classes: the variable-gradient method, which posits \(\nabla V\) with unknown coefficients and forces \(\dot V\) negative; Krasovskii's method, which tries \(V=\mathbf{f}^{T}\mathbf{f}\); and for mechanical systems, the physical energy itself, which is very often the right answer.

For a linear system the search is completely solved. Take the quadratic form \(V=\mathbf{x}^{T}P\mathbf{x}\) with \(P\) symmetric positive definite, and evaluate its derivative along \(\dot{\mathbf{x}}=A\mathbf{x}\):

The logic then runs backwards, which is what makes it useful. Choose any positive definite \(Q\) — \(Q=I\) is the standard choice — and solve the linear equations for \(P\). The theorem states that \(A\) has all its eigenvalues in the left half-plane if and only if the resulting \(P\) is positive definite. For linear systems, therefore, Lyapunov's method is not merely sufficient but exact, and it provides a stability test that requires no characteristic polynomial at all — only the solution of \(n(n+1)/2\) linear equations and a positive-definiteness check by Sylvester's criterion. Example 6 works one through.

The same machinery estimates a region of attraction, which is something nothing else in this book can do. If \(\dot V<0\) only inside some bounded region \(\Omega\), then the largest level set \(\{V(\mathbf{x})\le c\}\) that fits entirely inside \(\Omega\) is guaranteed to be attracted to the origin. That level set is a rigorous, if conservative, answer to the question the linearisation of Section 29-3 refused to answer: how large a disturbance can this system survive?

The describing function predicts; Lyapunov proves but demands ingenuity. Between them sits a class of results that prove stability for an entire family of nonlinearities at once, using only a frequency-domain test on the linear part. They address the Lur'e problem of Section 29-2 with the nonlinearity confined to a sector: \(k_1x^2\le x f(x)\le k_2x^2\), which says only that the characteristic stays between two straight lines through the origin. Nothing else about \(f\) is assumed — not continuity, not differentiability, not even that it is the same function from one instant to the next.

The circle criterion is the exact generalisation of Nyquist: shrink the sector to a single gain \(k_1=k_2=k\) and the disc collapses to the point \(-1/k\), recovering Chapter 16 word for word. The Popov criterion is the sharper of the two because it exploits time-invariance, and it has a graphical form as neat as a Bode plot — plot \(\operatorname{Re}G(j\omega)\) against \(\omega\operatorname{Im}G(j\omega)\), and the condition is that this Popov plot lie entirely to the right of a straight line of slope \(1/q\) through the point \(-1/k\). Both criteria are sufficient only, and both are conservative: a system can fail them and still be perfectly stable. Their value is that they are guarantees, and a guarantee for a whole sector covers all the modelling uncertainty in the nonlinearity at once.

It is worth recording why these results were needed. Aizerman conjectured that a system whose nonlinearity lies in a sector for which every constant gain would be stabilising must itself be stable. The conjecture is intuitive, widely believed for years, and false — counterexamples exhibit limit cycles. The sector criteria are what replaced it.

Design methods that deliberately exploit nonlinearity fall into a few families, each of which the reader will meet again in more depth elsewhere.

Sliding mode deserves a closer look, because Example 2 has already drawn it. Define a switching function \(s(\mathbf{x})=\dot x+\lambda x\) and choose \(u=-M\,\text{sgn}(s)\). Wherever \(s\dot s<0\) the control drives the state toward the surface \(s=0\), and once there the switching is infinitely fast and the state is pinned to the surface. On the surface \(\dot x=-\lambda x\), so the motion decays as a first-order system with time constant \(1/\lambda\) — governed entirely by the designer's choice of \(\lambda\) and completely independent of the plant. That insensitivity to plant parameters is the point of the method, and it makes sliding mode the most robust controller in this book. Its defect is equally sharp: real switching is not infinitely fast, so the state chatters about the surface at whatever frequency the hardware permits, exciting unmodelled dynamics. Replacing the sign function by a saturation of small width — a boundary layer — trades a little of the robustness for a great deal of smoothness.

For every describing-function problem, check that the linear part is low-pass at the predicted frequency before trusting the answer. Difficulty rises down the list.