2.1 States, Inputs, Outputs, and Disturbances

Assign each quantity one role¶

A dynamic model is useful for CCD only when every quantity has a clear engineering role. Consider

\dot{x}=f(t,x,u,d,p), \qquad y=h(t,x,u,d,p),

where $x(t)$ is the state, $u(t)$ is the manipulated input, $d(t)$ is an environmental or exogenous disturbance, $y(t)$ is an output, and $p$ is a vector of time-independent physical design variables.

The state is the minimum information needed, together with future inputs and disturbances, to predict future behavior. States usually represent stored energy or memory: position and momentum, capacitor charge, temperature, fluid inventory, or estimator memory. An input is a command the controller can choose. A disturbance influences the system but is not chosen by the controller. An output is a quantity calculated or measured for feedback, constraints, or performance evaluation.

Mass-spring-damper example¶

For a mass $m$ , spring $k$ , damper $c$ , actuator force $u$ , and base displacement $r$ , let $q$ be mass displacement. One model is

m\ddot q+c(\dot q-\dot r)+k(q-r)=u.

With $x=[q,\dot q]^T$ and disturbance $d=[r,\dot r]^T$ ,

\dot{x}=\begin{bmatrix}0&1\\-k/m&-c/m\end{bmatrix}x +\begin{bmatrix}0\\1/m\end{bmatrix}u +\begin{bmatrix}0&0\\k/m&c/m\end{bmatrix}d.

If the design variables are $p=[m,k,c,F_{\max}]^T$ , then the matrices and the constraint $|u|\le F_{\max}$ both depend on physical design.

Measured outputs are not performance outputs¶

A displacement sensor may provide $y_m=q-r$ , while the objective penalizes acceleration $y_p=\ddot q$ . The first is available to the controller; the second may be computed only for evaluation. Confusing them silently grants the controller information it does not possess.

Similarly, a road profile may be a disturbance. If a preview sensor measures it ahead of the vehicle, some portion becomes an available feedforward signal. Information availability changes the controller class and can change the optimal plant.

State choice and design dependence¶

Different state coordinates describe the same dynamics, but some expose design dependence more clearly. Modal coordinates are useful for flexible structures, energy variables for multiphysics systems, and relative coordinates for suspensions. A model developed around fixed numerical parameters may be excellent for control tuning yet unusable for CCD if it cannot be reassembled when geometry or material variables change.

Activity 2.1: classify the variables¶

Activity 2.1: classify the variables

An electric-vehicle battery and its coolant are represented by the two-lump model

C_b\dot T_b=Q_g-UA(v)(T_b-T_c),

C_c\dot T_c=UA(v)(T_b-T_c)-\dot m c_p(T_c-T_a),

with algebraic relations

UA(v)=U_0+k_vv, \qquad \dot m=k_mD_p\omega_p, \qquad P_p=\alpha D_p\omega_p^3.

The measured and performance outputs are

y_m=T_c+n_T, \qquad y_p=\begin{bmatrix}T_b & P_p\end{bmatrix}^{\mathsf T}.

Use the operating data below:

Quantity	Value	Quantity	Value
$C_b$	180 kJ/K	$C_c$	25 kJ/K
$c_p$	3,800 J/(kg K)	$U_0$	350 W/K
$k_v$	8 W s/(K m)	$k_m$	0.20 kg/s
$\alpha$	400 W	$D_p$	1.20
$T_b$	42°C	$T_c$	32°C
$T_a$	25°C	$Q_g$	5.0 kW
$v$	20 m/s	$\omega_p$	0.70
$n_T$	0.15 K	$T_{\mathrm{ref}}$	35°C

Construct a classification table for $T_b,T_c,\omega_p,Q_g,T_a,v,D_p,\dot m,P_p,n_T,T_{\mathrm{ref}},y_m,$ and $y_p$ . Label each as a state, manipulated input, disturbance, design variable, fixed parameter, algebraic variable, reference, or output. A quantity may have a primary and secondary label.
Write explicit vectors $x$ , $u$ , $d$ , and $p$ , and state which components of $y_p$ are actually available to the controller when only the coolant-temperature sensor is installed.
At the stated operating point, calculate $UA$ , $\dot m$ , $P_p$ , $\dot T_b$ , $\dot T_c$ , and $y_m$ . Report temperature rates in K/min.
Determine whether the feedback law $\omega_p=0.08(T_b-T_{\mathrm{ref}})$ is implementable with the stated sensor set. If not, identify the missing measurement or estimator output.
Replace the algebraic flow relation with pump dynamics,

\tau_p\frac{d\dot m}{dt}+\dot m=k_mD_p\omega_p, \qquad \tau_p=0.40\ \mathrm{s}.

State the new state vector and the additional initial condition required. Explain quantitatively why treating $\dot m$ as algebraic may be questionable for commands varying on a time scale comparable to 0.40 s. 6. Explain how the role of vehicle speed $v$ changes if it is prescribed by a drive cycle versus commanded by a supervisory controller.