State-Space Representation

State-Space Representation

A state-space model describes a dynamical system as a set of first-order ordinary differential equations in a state vector $x(t)\in {\mathbb{R}}^n$, driven by an input $u(t)\in {\mathbb{R}}^m$ and producing an output $y(t)\in {\mathbb{R}}^p$:
$\dot{x}(t)=f(x(t),u(t),t),y(t)=h(x(t),u(t),t).$
The state is the minimal set of variables such that knowing $x(t_0)$ and $u(t)$ for $t\ge t_0$ uniquely determines $x(t)$ and $y(t)$ for all future time. The choice of states is not unique, but the number of states $n$ (the order of the system) is fixed.

When $f$ and $h$ are linear in $(x,u)$ and their coefficients are time-invariant, the system is linear time-invariant (LTI) and reduces to four constant matrices:
$\dot{x}(t)=Ax(t)+Bu(t),y(t)=Cx(t)+Du(t),$
with $A\in {\mathbb{R}}^{n\times n}$ (the system matrix or state matrix), $B\in {\mathbb{R}}^{n\times m}$ (input matrix), $C\in {\mathbb{R}}^{p\times n}$ (output matrix), and $D\in {\mathbb{R}}^{p\times m}$ (feedthrough matrix, usually zero for physical systems). The LTI form is the workhorse of linear control theory — every tool in this section depends on it.

For a robot, the state vector typically contains configuration and velocity: $x=(\theta ,\dot{\theta })$ for an $n$-joint manipulator has $2n$ states. The input $u$ is usually joint torques, and the output $y$ is whatever the sensors measure (joint angles, end-effector pose, forces).

Dynamic vs Kinematic State Dimension $6$ kinematic states ($\theta \in {\mathbb{R}}^6$) but $12$ dynamic states ($(\theta ,\dot{\theta })\in {\mathbb{R}}^{12}$). The manipulator dynamics are second-order in $\theta$, so converting to state-space doubles the state dimension. Mixing up the two is a common source of off-by-a-factor-of-two bugs in controller dimension.

A 6-DoF arm has

Intuition

The state vector is the minimum information the system has to “remember” to predict its future. For a point mass, position alone is not enough — you also need velocity, because the future path depends on how fast it is moving. For a second-order mechanical system, (position, velocity) is the standard state; for an RLC circuit, (inductor current, capacitor voltage) is standard. The same physical system can have many valid state vectors; what matters is that the chosen set is complete (no essential variable missing) and minimal (no redundant variable).

Writing the dynamics as first-order ODEs, rather than higher-order ODEs, matters because every tool in linear systems theory — eigenvalue analysis, the state transition matrix, controllability tests — assumes first-order form. Converting a second-order ODE $\ddot{y}+a\dot{y}+by=u$ to first-order form is a routine move: introduce $x_1=y$, $x_2=\dot{y}$, and write ${\dot{x}}_1=x_2$, ${\dot{x}}_2=-b{x}_1-a{x}_2+u$.

Properties

Structure

(First-order form) State-space always uses first-order ODEs; higher-order equations are converted by introducing velocity-like variables.

A scalar $n$-th order ODE $y^{(n)}=f(y,\dot{y},\dots ,y^{(n-1)},u)$ becomes $n$ first-order equations with state $x=(y,\dot{y},\dots ,y^{(n-1)})$. This is why a second-order mechanical system needs $2n$ states to represent $n$ generalized coordinates.

(Minimality) The state vector must contain every variable needed to predict the future, and no more.

“Need to predict” is defined precisely: $(x(t_0),u(t\ge t_0))$ determines $(x(t),y(t))$ uniquely. Adding redundant variables violates minimality; removing essential variables violates the predictive property. A minimal state is not unique — any invertible linear transformation $\tilde{x}=Tx$ gives a valid alternative.

(Input, state, output) Inputs are applied, states are internal, outputs are measured.

Inputs $u$ are signals the controller can set (motor torques, voltages). States $x$ are the internal variables describing the system; some may be directly observable, others estimated. Outputs $y$ are the subset of information actually measured by sensors. In textbook problems $y=x$ often; in real robots $y\ll x$ because not every state has a sensor.

LTI Form

(Superposition) Solutions to LTI systems scale and add linearly.

If $x_1(t)$ is the response to $(x_1(0),u_1)$ and $x_2(t)$ is the response to $(x_2(0),u_2)$, then $\alpha x_1+\beta x_2$ is the response to $(\alpha x_1(0)+\beta x_2(0),\alpha u_1+\beta u_2)$. Superposition is the reason LTI systems admit Fourier, Laplace, and convolution analysis.

(Time invariance) If $u(t)$ produces $x(t)$, then $u(t-\Delta )$ produces $x(t-\Delta )$.

The matrices $A$, $B$, $C$, $D$ are constant in time. Time-varying systems $\dot{x}=A(t)x+B(t)u$ arise when system parameters change (mass depleting on a rocket, joint configuration changing the effective inertia of a nonlinear manipulator around an operating trajectory), but LTI is the starting point.

(Feedthrough) $D\ne 0$ only when the input directly influences the output without passing through the state.

For almost all physical systems, outputs depend on states (positions, voltages) rather than directly on inputs (torques), so $D=0$. Nonzero $D$ arises in some electrical networks and in systems with algebraic constraints.

Discrete Time

(Difference equation form) Discrete-time state-space replaces the derivative with a one-step shift: $x[i+1]=A_{d}x[i]+B_{d}u[i]$.

Sampling a continuous-time LTI system at intervals $T$ produces a discrete-time LTI system with $A_d=e^{AT}$ and $B_d={\int }_0^T{e}^{A\tau }Bd\tau$. Every continuous-time tool has a discrete-time analogue, often with modified eigenvalue conditions (see 7.5. Stability).

Robotics Context

(Manipulator state) The state of an $n$-DoF manipulator is $(\theta ,\dot{\theta })\in {\mathbb{R}}^{2n}$ with input $u={\rm Y}\in {\mathbb{R}}^n$.

Rewriting the manipulator equation ${\rm Y}=M(\theta )\ddot{\theta }+C(\theta ,\dot{\theta })\dot{\theta }+G(\theta )$ as a first-order system,
$\left[\begin{array}{c}\dot{\theta }\\ \ddot{\theta }\end{array}\right]=\left[\begin{array}{c}\dot{\theta }\\ M^{-1}({\rm Y}-C\dot{\theta }-G)\end{array}\right].$
The result is nonlinear (because of $M(\theta )$, $C(\theta ,\dot{\theta })$, $G(\theta )$) but first-order in $(\theta ,\dot{\theta })$. Linearization about an equilibrium pose produces an LTI model suitable for controller design — see 7.2. Linearization.

Examples

Mass-Spring System

A mass $m$ on a spring of stiffness $k$ with position $y$ obeys $m\ddot{y}+ky=F$. Write this in state-space form with $F$ as input and $y$ as output.

Solution $x_1=y$, $x_2=\dot{y}$. Then ${\dot{x}}_1=x_2$ and ${\dot{x}}_2=-(k/m)x_1+(1/m)F$.

Define

$A=\left[\begin{array}{cc}0& 1\\ -k/m& 0\end{array}\right],B=\left[\begin{array}{c}0\\ 1/m\end{array}\right],C=\left[\begin{array}{cc}1& 0\end{array}\right],D=0.$

Order $n=2$, one input, one output.

1-DoF Revolute Joint with Gravity

A single link of mass $m$ and length $\ell$ hangs from a pivot with joint angle $\theta$ from vertical, driven by torque $\tau$. With moment of inertia $I=\frac{1}{3}m{\ell }^2$ about the pivot, the dynamics are $I\ddot{\theta }=\tau -\frac{1}{2}mg\ell \sin \theta$. Write the state-space model.

Solution $x=(\theta ,\dot{\theta })$, input $u=\tau$.

State

${\dot{x}}_1=x_2$

${\dot{x}}_2=(1/I)(u-\frac{1}{2}mg\ell \sin x_1)$

This is nonlinear because of $\sin x_1$. Linearizing about the downward equilibrium ($x=0$, $u=0$) replaces $\sin x_1\approx x_1$ and gives an LTI system with $A=\left[\begin{array}{cc}0& 1\\ -mg\ell /(2I)& 0\end{array}\right]$ and $B=\left[\begin{array}{c}0\\ 1/I\end{array}\right]$.

See Also

• 7.2. Linearization — converting $f$, $h$ to $(A,B,C,D)$ via Taylor expansion
• 7.3. Solving LTI Systems — closed-form $x(t)=e^{At}x(0)+\int e^{A(t-\tau )}Bu(\tau )d\tau$
• 6.3. Manipulator Equations of Motion — the source of the robot’s nonlinear state equations