Pole Placement, State Feedback, and Observer Design

Every design method in Parts 3, 4 and 5 shared a quiet limitation. The root locus of Chapter 12 showed where the closed-loop poles can go as a single gain \(K\) varies, and Chapter 13 taught us to pick a point on that locus. The locus itself was fixed by the plant; if the point we wanted was not on it, we had to bend the locus by adding a compensator pole and zero (Chapter 21) and try again. Frequency-domain design in Chapters 17 and 22 worked the same way: one scalar gain, one phase-lead network, an iterative search.

The reason for that limitation is structural. Output feedback \(u = -k\,y\) uses a single measured signal, so it offers a single number to choose. But Chapter 23 taught us that the internal condition of an \(n\)th-order system is an \(n\)-vector \(\mathbf{x}\), and Chapter 24 showed that \(\mathbf{x}(t)\) carries everything there is to know about the system's future. If we could feed back all of \(\mathbf{x}\), each of its \(n\) components would come with its own gain — \(n\) numbers to choose, and \(n\) closed-loop poles to place. That count is not a coincidence, and this chapter turns it into a design method.

Two obstacles stand in the way, and Chapter 25 has already named both. Feeding back a state is useless if the input cannot influence that state, which is the question of controllability. And feeding back a state is impossible if no sensor measures it, which is answered by observability — if the missing states are observable we can compute them from the output history rather than measure them. Sections 26-2 through 26-7 solve the first problem; Sections 26-8 and 26-9 solve the second.

Take the single-input state model of Chapter 23,

with \(\mathbf{A}\) an \(n\times n\) matrix and \(\mathbf{B}\) an \(n\times 1\) column. Suppose for the moment that every component of \(\mathbf{x}\) is available for measurement. The state feedback law forms the input as a weighted sum of all the states:

The row vector \(\mathbf{K}\) is the state feedback gain matrix; the minus sign is convention, so that positive \(k_i\) means negative feedback. Substituting into the plant collapses the loop into a single autonomous equation:

Read the result carefully, because two things did not change. The input matrix \(\mathbf{B}\) is untouched, and so is the output matrix \(\mathbf{C}\). Since the transfer-function numerator of Chapter 23 is built from \(\mathbf{C}\,\mathrm{adj}(s\mathbf{I}-\mathbf{A})\,\mathbf{B}\), and one can show the adjugate combination is unaffected by the substitution \(\mathbf{A}\to\mathbf{A}-\mathbf{BK}\), state feedback moves the poles but leaves the zeros exactly where they were. Section 26-7 returns to the consequences.

The figure below shows the arrangement. A scalar reference is admitted through a feedforward gain \(\bar{N}\) whose purpose is explained in Section 26-7; for pure regulation (driving the state to zero from a disturbed initial condition) that path is absent and \(r=0\).

The design problem now states itself. We are handed a set of \(n\) desired closed-loop pole locations \(\mu_1,\dots,\mu_n\) — chosen from a transient specification exactly as in Chapter 8, and closed under complex conjugation so that \(\mathbf{K}\) comes out real. Form the desired characteristic polynomial

and require the actual closed-loop polynomial to equal it:

Both sides are monic polynomials of degree \(n\), so matching them means matching \(n\) coefficients. Because \(\mathbf{K}\) enters \(\mathbf{A}-\mathbf{BK}\) linearly, and the determinant is linear in each row, every coefficient of the left-hand polynomial is a linear function of \(k_1,\dots,k_n\). We therefore face \(n\) linear equations in \(n\) unknowns — a system that has a unique solution precisely when its coefficient matrix is nonsingular, and that nonsingularity turns out to be the controllability condition of Chapter 25.

For second- and third-order systems this is the fastest route by hand: expand the determinant symbolically and equate coefficients. Take a double integrator with drag, \(\mathbf{A}=\begin{bmatrix}0&1\\0&-2\end{bmatrix}\), \(\mathbf{B}=\begin{bmatrix}0\\1\end{bmatrix}\), and \(\mathbf{K}=\begin{bmatrix}k_1&k_2\end{bmatrix}\). Then

Every coefficient is now a knob: \(k_1\) sets the constant term outright and \(k_2\) sets the \(s\)-coefficient. Any polynomial \(s^2+\alpha_1 s+\alpha_0\) can be produced, so any pair of poles can be reached. That freedom is complete only because the two unknowns appeared in two different coefficients; when a system is uncontrollable, one or more coefficients lose all dependence on \(\mathbf{K}\) and the corresponding modes are frozen.

Coefficient matching by hand becomes unpleasant beyond third order. The systematic route exploits a coordinate system in which the answer is written down by inspection. Recall from Chapter 23 the controllable canonical form of a system whose open-loop characteristic polynomial is \(\Delta(s) = s^n + a_{n-1}s^{n-1}+\cdots+a_0\):

The whole characteristic polynomial lives in the bottom row, and \(\bar{\mathbf{B}}\) touches only that row. So if we feed back \(\bar{\mathbf{u}} = -\bar{\mathbf{K}}\bar{\mathbf{x}}\) with \(\bar{\mathbf{K}}=\begin{bmatrix}\bar{k}_1&\cdots&\bar{k}_n\end{bmatrix}\), the matrix \(\bar{\mathbf{A}}-\bar{\mathbf{B}}\bar{\mathbf{K}}\) is identical to \(\bar{\mathbf{A}}\) except that its bottom row becomes \(\begin{bmatrix}-(a_0+\bar{k}_1) & -(a_1+\bar{k}_2) & \cdots & -(a_{n-1}+\bar{k}_n)\end{bmatrix}\). It is still in canonical form, so its characteristic polynomial can be read straight off:

A general plant is not handed to us in that form, but Chapter 25 showed how to get there. The transformation \(\mathbf{x}=\mathbf{T}\bar{\mathbf{x}}\) with \(\mathbf{T}=\mathbf{M}_c\mathbf{W}\) carries the canonical pair into the given one, where \(\mathbf{M}_c\) is the controllability matrix and \(\mathbf{W}\) is the Hankel matrix of the open-loop coefficients:

Because \(u=-\bar{\mathbf{K}}\bar{\mathbf{x}} = -\bar{\mathbf{K}}\mathbf{T}^{-1}\mathbf{x}\), the gain in the original coordinates is \(\mathbf{K}=\bar{\mathbf{K}}\mathbf{T}^{-1}\). Written out, this is the Bass–Gura formula:

The formula makes the controllability requirement visible: \(\mathbf{T}^{-1}\) exists only if \(\mathbf{M}_c\) is nonsingular, since \(\mathbf{W}\) is always nonsingular (its anti-diagonal is all ones). An uncontrollable plant simply has no such coordinate change, and the canonical form does not exist.

Bass–Gura works but asks for two matrices. Ackermann's formula compresses the whole construction into one line, and it is the version worth memorising for examinations.

The formula looks unmotivated until one asks what \(\phi(\mathbf{A})\) actually measures. By the Cayley–Hamilton theorem of Chapter 24, every matrix satisfies its own characteristic equation, so \(\Delta(\mathbf{A}) = \mathbf{A}^n + a_{n-1}\mathbf{A}^{n-1}+\cdots+a_0\mathbf{I} = \mathbf{0}\). Subtracting this zero from \(\phi(\mathbf{A})\) costs nothing and reveals the structure:

The \(n\)th-power terms cancel exactly, and what survives is a polynomial in \(\mathbf{A}\) of degree \(n-1\) whose coefficients are precisely the differences \(\alpha_i - a_i\) that Section 26-4 identified as the canonical gain. The premultiplier \(\begin{bmatrix}0&\cdots&0&1\end{bmatrix}\mathbf{M}_c^{-1}\) is the row that extracts those coefficients and converts them back to the original coordinates; carrying out that verification in canonical form reproduces \(\bar{\mathbf{K}}\) exactly. So Ackermann's formula is Bass–Gura with the bookkeeping done in advance.

Two practical remarks. First, \(\mathbf{M}_c^{-1}\) is required, so once again controllability is the gatekeeper — and a nearly uncontrollable plant gives a badly conditioned \(\mathbf{M}_c\), producing enormous and unreliable gains. Second, for \(n\) beyond four or five the formula is numerically poor and production software uses orthogonal-transformation algorithms instead; for hand calculation up to third order it is unbeatable.

The algebra will place the poles anywhere we ask, which makes the asking the hard part of the design. Nothing in the formulas penalises a foolish choice, so the engineer must supply the judgement.

The usual construction begins with the transient specification. From Chapter 8, an overshoot and a settling time fix a damping ratio and a natural frequency,

and the remaining \(n-2\) poles are placed far enough to the left that they contribute negligibly — the dominant-pole idea of Chapter 8, now enforced by construction instead of hoped for. A common rule is to put them five to ten times further left than the real part of the dominant pair, with the extra poles kept real to avoid introducing new oscillatory modes.

The temptation is to place all the poles very far left and enjoy a very fast response. That temptation must be resisted, for four reasons that no formula will remind you of.

Section 26-2 noted that \(\mathbf{B}\) and \(\mathbf{C}\) survive the feedback untouched. The consequences deserve to be spelled out, because a design that places the poles perfectly can still track the reference badly.

The zeros do not move. Whatever \(\mathbf{K}\) we choose, the closed-loop transfer function \(\mathbf{C}(s\mathbf{I}-\mathbf{A}+\mathbf{BK})^{-1}\mathbf{B}\) has the same numerator as the open-loop plant. A non-minimum-phase zero in the right half-plane — the source of the initial undershoot discussed in Chapter 6 — is still there afterwards, and the fundamental limitation it imposes on achievable bandwidth is untouched. Pole placement is not a cure for bad plant structure.

The DC gain is wrong. Setting \(u=-\mathbf{Kx}\) drives the state to zero, which is exactly what a regulator should do but useless for tracking. Admitting the reference as \(u=-\mathbf{Kx}+\bar{N}r\) gives a steady state \(\mathbf{x}_{ss} = -(\mathbf{A}-\mathbf{BK})^{-1}\mathbf{B}\bar{N}r\), so \(y_{ss}=r\) requires

And that scaling is fragile. \(\bar{N}\) is computed from the model. If the plant gain drifts by ten per cent, the output is ten per cent wrong for ever, and no amount of pole placement notices — the loop has no integrator, so by the type analysis of Chapter 9 it is a type-0 system with a finite step error. A constant load disturbance produces a permanent offset for the same reason.

The repair is the one Chapter 19 taught: put an integrator in the loop, and put it there before designing the gains. Augment the state with the integral of the tracking error, \(\dot{x}_I = r-y = r-\mathbf{Cx}\), and design for the enlarged \((n+1)\)th-order plant

Now \(n+1\) poles are placed, the extra one associated with the integrator, and the closed loop is type 1: constant references are tracked and constant disturbances rejected with zero steady-state error, robustly, without depending on the accuracy of any feedforward constant. The augmented pair \((\mathbf{A}_a,\mathbf{B}_a)\) is controllable provided the original pair is controllable and the plant has no zero at the origin.

Everything so far assumed that all \(n\) states can be measured. In practice they cannot. A servomotor has an encoder on the shaft but no tachometer; an aircraft measures attitude but not every aerodynamic state; a chemical reactor measures temperature but not concentration. Instrumenting every state is expensive, sometimes impossible, and always one more thing to fail.

The way out was found by Luenberger in 1964, and it is beautifully direct: build a simulation of the plant inside the controller, run it in parallel with the real machine, and correct it whenever its predicted output disagrees with the measured one. Call the simulated state \(\hat{\mathbf{x}}\). A naive copy would be \(\dot{\hat{\mathbf{x}}}=\mathbf{A}\hat{\mathbf{x}}+\mathbf{B}u\), which drifts away from the truth for ever if the initial condition is wrong. Adding the correction term gives the full-order observer:

To see whether the estimate converges, define the estimation error \(\mathbf{e}=\mathbf{x}-\hat{\mathbf{x}}\) and subtract the observer equation from the plant equation. The input \(u\) is fed to both, so it cancels — the estimator does not need to know what the controller is doing, only what it did:

The estimation error obeys a homogeneous linear equation of its own. If every eigenvalue of \(\mathbf{A}-\mathbf{LC}\) lies in the left half-plane, then \(\mathbf{e}(t)\to\mathbf{0}\) from any starting mismatch, and how fast is entirely up to us — a design problem identical in form to the one we just solved for \(\mathbf{A}-\mathbf{BK}\).

That identity is not a coincidence but the duality of Chapter 25. Because a matrix and its transpose share eigenvalues, \(\det(s\mathbf{I}-\mathbf{A}+\mathbf{LC}) = \det(s\mathbf{I}-\mathbf{A}^T+\mathbf{C}^T\mathbf{L}^T)\). The right-hand side is a state feedback problem for the pair \((\mathbf{A}^T,\mathbf{C}^T)\) with gain \(\mathbf{L}^T\), and it is solvable for arbitrary poles exactly when \((\mathbf{A}^T,\mathbf{C}^T)\) is controllable — which is precisely the statement that \((\mathbf{A},\mathbf{C})\) is observable.

Every technique of Sections 26-3 to 26-5 therefore transfers verbatim: coefficient matching, canonical forms, and the dual Ackermann formula in the last row of the table.

Choosing the observer poles follows one clear rule and one quiet caution. The rule: the estimate must converge much faster than the controlled dynamics, or the controller will spend the transient acting on stale information — so place the observer poles two to six times further left than the dominant closed-loop poles. The caution: a fast observer means a large \(\mathbf{L}\), and \(\mathbf{L}\) multiplies the measurement, so it multiplies the measurement noise as well. The observer is a filter, and its bandwidth is a compromise between trusting the model and trusting the sensor. The optimal compromise, given the statistics of the noise, is the Kalman filter of Chapter 30 — structurally identical to this observer, with \(\mathbf{L}\) computed rather than chosen.

Two designs are now in hand: a gain \(\mathbf{K}\) that would work if we knew \(\mathbf{x}\), and an observer that produces \(\hat{\mathbf{x}}\). The obvious step is to combine them and feed back the estimate,

but the step is not obviously safe. The controller is now acting on an estimate that is itself wrong during the transient, and the plant's response feeds back into the very measurement the observer uses. Do the two designs interfere? The answer, remarkably, is no.

Write the closed loop in the coordinates \((\mathbf{x},\mathbf{e})\) with \(\mathbf{e}=\mathbf{x}-\hat{\mathbf{x}}\), so that \(\hat{\mathbf{x}} = \mathbf{x}-\mathbf{e}\). The plant equation becomes \(\dot{\mathbf{x}} = \mathbf{Ax}-\mathbf{BK}(\mathbf{x}-\mathbf{e}) = (\mathbf{A}-\mathbf{BK})\mathbf{x}+\mathbf{BKe}\), and the error equation we already derived is untouched by \(u\):

The zero block in the lower left is the whole content of the theorem: the estimation error drives the state, but the state does not drive the estimation error. That is why the observer could be designed in Section 26-8 without any knowledge of \(\mathbf{K}\).

Two cautions belong with the result. First, separation guarantees the pole locations, not the transient shape — the term \(\mathbf{BKe}\) does inject the estimation transient into the state response, which is exactly why the observer is made faster than the regulator. Second, separation is a statement about the nominal model; the stability margins of the combined loop can be much poorer than those of the ideal full-state design, a phenomenon known as loss of the guaranteed margins of state feedback, and one of the reasons robust design in Chapter 30 exists.

Finally, it is worth seeing what the finished controller is, in the language of Parts 3 to 5. Substituting \(u=-\mathbf{K}\hat{\mathbf{x}}\) and \(y=\mathbf{Cx}\) into the observer gives a self-contained dynamic system driven by the measurement alone:

This is an \(n\)th-order transfer function in the forward path of an ordinary unity-feedback loop — a cascade compensator of exactly the kind Chapter 21 built from resistors and capacitors, except that here it was derived from a specification rather than tuned into place. Example 6 computes one, and confirms that closing the loop with it reproduces the \(2n\) poles the separation principle promises.

One refinement is worth knowing. If \(y\) already measures one state directly, estimating that state again is wasted effort. The reduced-order (minimal) observer estimates only the \(n-m\) unmeasured combinations, giving a controller of order \(n-m\) instead of \(n\). It is cheaper and faster, but it feeds the measurement straight through to the estimate, so sensor noise reaches the control signal unfiltered — full-order observers are often preferred in noisy hardware for exactly that reason.

Test controllability or observability before designing anything; a rank deficiency changes the question rather than the arithmetic. Difficulty rises down the list.