State-Space Representation of Systems

Everything in the first five parts of this course has rested on one object: the ratio \(C(s)/R(s)\). It has been remarkably productive. From it came the poles that fix stability (Chapters 10 and 11), the locus that shows how those poles move with gain (Chapter 12), the frequency response that can be measured without ever knowing the model (Chapter 14), and the compensators of Part 5. It is worth pausing to notice how much that one ratio was made to carry — and then to be honest about what it cannot carry at all.

It is defined only for zero initial conditions. The derivation in Chapter 3 dropped every initial-condition term the Laplace transform produced. That is harmless when a system starts at rest, and useless when it does not — a satellite already tumbling, a furnace already hot, a motor already turning.

It sees only the two ends. A transfer function relates one input to one output. It says nothing about the armature current that flows to produce a perfectly acceptable shaft angle, nothing about the internal pressure in a tank whose level is being controlled well. Designs fail on internal variables that the input–output description never mentions.

It can lose modes entirely. When a numerator factor cancels a denominator factor, the corresponding mode disappears from \(C(s)/R(s)\) while remaining perfectly present in the physical system. If that mode happens to be unstable, the transfer function will pronounce a diverging system stable. Section 23-8 shows how this happens and Chapter 25 gives it its proper names.

It scales badly. A plant with three inputs and two outputs needs six transfer functions, and every manipulation becomes bookkeeping. Time-varying and nonlinear systems have no transfer function at all, so the entire classical toolbox simply stops.

The state-space description answers all four. It carries initial conditions naturally, it names every internal variable explicitly, it retains modes that cancel, and it handles many inputs and many outputs with no change of notation whatsoever. It is the language of everything that follows — the solution and the transition matrix in Chapter 24, controllability and observability in Chapter 25, pole placement and observers in Chapter 26, and the optimal and adaptive methods of Chapter 30.

The word "state" has a technical meaning that is worth stating carefully, because the whole subject follows from it.

Two words in that definition do the work. Smallest: any redundant variable can be computed from the others and so is not part of the state. Determined: the state is a complete summary of the past. Two systems started from the same state and driven by the same input behave identically thereafter, however different their histories were. This is why the state is sometimes described as the memory of the system: it is exactly what the past has to communicate to the future.

For a physical system the state variables have a concrete identity: they are the variables that describe independently stored energy. A capacitor stores \(\tfrac12 Cv_C^{2}\), an inductor \(\tfrac12 Li_L^{2}\), a moving mass \(\tfrac12 mv^{2}\), a stretched spring \(\tfrac12 kx^{2}\), a thermal mass \(mc\,\theta\). Each independent store contributes one state variable, and the natural choice is the variable in whose square the energy is written.

The word independent is not decoration. Two capacitors in parallel store energy jointly, not separately: their voltages are equal, so they contribute one state variable, not two. A loop consisting only of capacitors and voltage sources, or a node joining only inductors and current sources, similarly removes one degree of freedom. In such cases the count of components exceeds the order of the system, and it is the order that is right.

Once the state variables are chosen, the physics is written as a rate law for each of them. For a linear time-invariant system every rate is a linear combination of the states and the inputs, and every output is a linear combination of the states and the inputs. Two matrix equations say all of it.

An \(n\)th-order scalar differential equation has been traded for \(n\) coupled first-order equations, and the trade is a good one. First-order vector equations have a clean existence-and-uniqueness theory; they integrate numerically with standard routines; they generalise to time-varying coefficients by simply writing \(A(t)\); and, as Chapter 24 will show, they have a solution in closed form that looks exactly like the scalar solution \(x=e^{at}x_0\) with the scalar exponential replaced by a matrix exponential.

The block diagram of the model makes its structure visible. Each state variable is the output of an integrator; \(A\) is the feedback path around the bank of integrators; \(B\) injects the input; \(C\) reads the states out; \(D\) bypasses the dynamics entirely.

When a model of the physical system exists — the differential equations of Chapter 2 — the state equations follow by a fixed procedure. Choose one state variable per independent energy store; write the element law that gives the derivative of each; eliminate every variable that is not a state or an input; assemble.

Take the mass–spring–damper of Chapter 2, \(m\ddot q+b\dot q+kq=f\), with position \(q\) as the output. The spring stores energy in \(q\) and the mass in \(\dot q\), so \(n=2\) and the natural choice is \(x_1=q\), \(x_2=\dot q\). The first rate law is a definition; the second is Newton's law rearranged.

Electrical systems follow the same pattern with Kirchhoff's laws in place of Newton's. The rule of thumb is to write the inductor law \(L\,di_L/dt=v_L\) and the capacitor law \(C\,dv_C/dt=i_C\), then use the loop and node equations to express \(v_L\) and \(i_C\) in terms of the states and the sources. Example 1 carries this out for the series RLC circuit below.

Often no physical model is available and the starting point is a differential equation or a transfer function. There is then no physics to guide the choice of state variables, and the standard trick is to take the output and its successive derivatives. These are the phase variables. Begin with the simplest case, an equation with no derivatives of the input:

Only the last equation carries any content; the rest are definitions. Assembled, the matrices take the companion shape: ones on the superdiagonal, the negated coefficients along the bottom row, and the input entering only the last state.

Numerator dynamics need one extra step. If the transfer function is \(Y(s)/U(s)=N(s)/D(s)\) with \(N(s)=b_ms^{m}+\cdots+b_0\) and \(m<n\), introduce an intermediate variable \(Z(s)\) defined by \(D(s)Z(s)=U(s)\). Then \(Y(s)=N(s)Z(s)\), and the phase variables are taken on \(z\) rather than on \(y\):

The matrices \(A\) and \(B\) are untouched: the denominator alone sets the dynamics, and the numerator appears entirely in \(C\). If \(m=n\), long division first produces a constant that becomes \(D\), leaving a strictly proper remainder to be handled as above. The simulation diagram of this form is a chain of integrators with the coefficients wired back to the input end and the numerator coefficients tapped forward — the state-space cousin of the signal flow graphs of Chapter 5.

The controllable form is one of infinitely many valid state models for a given transfer function, and two others are used constantly enough to be worth memorising.

The observable canonical form is the exact dual of the controllable one. Take the same transfer function, transpose \(A\), and swap the roles of \(B\) and \(C\):

That this transposition leaves the transfer function unchanged is easy to prove: a \(1\times1\) matrix equals its own transpose, so \(T(s)=C(sI-A)^{-1}B=\left[C(sI-A)^{-1}B\right]^{\mathsf T}=B^{\mathsf T}(sI-A^{\mathsf T})^{-1}C^{\mathsf T}\), which is precisely the observable form. The name comes from Chapter 25: this arrangement makes the states maximally visible from the output, just as the controllable form makes them maximally reachable from the input.

The diagonal or modal form is the most illuminating of the three. Expand the transfer function in partial fractions. If the poles \(\lambda_1,\ldots,\lambda_n\) are distinct,

Here the state equations decouple completely: \(\dot x_i=\lambda_ix_i+u\), one independent first-order system per mode. Nothing in the model couples the modes to one another; they merely add at the output. This makes the diagonal form the natural place to see stability (every \(\lambda_i\) must have a negative real part), and the natural place to see whether a mode can be influenced at all — if a row of \(B\) is zero the corresponding mode cannot be driven, and if a column of \(C\) is zero it cannot be seen.

Repeated poles spoil the diagonalisation. A pole of multiplicity \(k\) contributes a \(k\times k\) Jordan block, diagonal with the repeated eigenvalue and carrying ones on the superdiagonal, and the corresponding modes are coupled in a chain — which is the state-space fingerprint of the \(te^{\lambda t}\) terms that repeated poles produce in the time response.

Given \(A,B,C,D\), the transfer function follows in three lines. Laplace-transform the state equation, taking zero initial conditions since that is the transfer function's defining assumption:

Two consequences deserve attention. First, the roots of \(|sI-A|=0\) are by definition the eigenvalues of \(A\), so the poles of a system are the eigenvalues of its state matrix. Everything Chapters 10 and 11 said about pole locations transfers verbatim: the system is stable if and only if every eigenvalue of \(A\) lies in the open left half-plane.

Second, the poles of \(T(s)\) are a subset of the eigenvalues of \(A\), not necessarily all of them. If a factor of \(C\,\mathrm{adj}(sI-A)\,B\) cancels a factor of \(|sI-A|\), that eigenvalue vanishes from the transfer function while remaining an eigenvalue of \(A\) and a genuine mode of the system. This is the "lost mode" promised in Section 23-1, and it is examined next.

Keeping the initial conditions instead of discarding them gives \(\mathbf{X}(s)=(sI-A)^{-1}\left[\mathbf{x}(0)+B\mathbf{U}(s)\right]\), in which the same matrix \((sI-A)^{-1}\) propagates both the stored energy and the applied input. Its inverse transform is the state transition matrix, and it is the entire subject of Chapter 24.

Section 23-2 claimed that the state vector is not unique. Make that precise. Let \(P\) be any invertible \(n\times n\) matrix and define a new state vector by \(\mathbf{x}=P\mathbf{z}\). Substituting into the model gives a model of exactly the same form in the new variables:

The two models describe the same system in different coordinates. What survives the change? The determinant of a product is the product of the determinants, so

The eigenvalues are therefore untouched by any relabelling of the states, and so is the transfer function, since \(\bar C(sI-\bar A)^{-1}\bar B=CP\,P^{-1}(sI-A)^{-1}P\,P^{-1}B=C(sI-A)^{-1}B\). What changes is only the interpretation of the individual numbers in \(\mathbf{x}\). The matrices are coordinates; the eigenvalues are physics.

The most useful choice of \(P\) is the modal matrix, whose columns are the eigenvectors of \(A\). If the eigenvalues are distinct the eigenvectors are independent, \(P\) is invertible, and \(P^{-1}AP\) is diagonal with the eigenvalues on it — the diagonal form of Section 23-6, reached by transformation instead of by partial fractions. Work it through on a small example.

With \(B=[0\;\;1]^{\mathsf T}\) and \(C=[1\;\;0]\) the transformed model has \(\bar B=P^{-1}B=[1\;\;{-1}]^{\mathsf T}\) and \(\bar C=CP=[1\;\;1]\), and the transfer function computed in the new coordinates is \(\dfrac{1}{s+1}-\dfrac{1}{s+2}=\dfrac{1}{(s+1)(s+2)}\) — the same as before, as it must be.

Now the hidden-mode question. Consider the model

The system is second order and has eigenvalues \(-1\) and \(-2\), yet its transfer function is first order. The second row of \(B\) is zero, so the input cannot reach \(x_2\) at all: whatever \(x_2\) does, it does on its own, decaying from its initial value with no possibility of being influenced. The transfer function, which assumes zero initial conditions and describes only the input–output path, has no reason to mention it — so it does not. Reverse the situation by zeroing a column of \(C\) and the mode becomes invisible to the sensor instead. Both cases are what Chapter 25 will call loss of controllability and loss of observability, and both are why the order of a state model can exceed the order of its transfer function.

State the choice of state variables explicitly in every case, and check each answer by computing \(C(sI-A)^{-1}B+D\) and comparing with the transfer function you started from or expected.