7.4: Exercises

Exercise 7.1

Consider the nonlinear difference equation

\[y(k+n)=F[y(k+n-1), \ldots, y(k), u(k+n-1), \ldots, u(k), k]\nonumber\]

where \(n\) is a fixed integer, and \(k\) is the time index

(a) Find a state-space representation of order \(2n - 1\) for this difference equation

(b) Find an \(n\)th-order state-space representation in LTI case (what is the form of \(F\) in this case?), using z-transforms for guidance (natural state variables are the coefficients of the initial-condition terms in the z-transformed version of the difference equation - try a third-order difference equation - remind of forward shift theorem from z-transforms). This part will guide the solution of (c).

(c) Find an \(n\)th-order state-space representation for the nonlinear system in (a) for the case where \(F\) [ . ] has the special form

\[F[.]=\sum_{i=1}^{n} f_{i}[y(k+n-i), u(k+n-i)]\nonumber\]

(Hint: Note that the difference equation in part (b) has this form; use your definition of state variables in (b) to guide your choice here.)

Exercise 7.2

Consider a causal continuous-time system with input-output representation \(y(t) = h * u(t)\), where * denotes convolution and \(h(t)\) is the impulse response of the system:

\[h(t)=2 e^{-t}-c e^{-2 t} \quad \text { for } t \geq 0\nonumber\]

Here \(c\) denotes a constant.

(a) Suppose \(c = 2\). Use only the input-output representation of the system to show that the variables \(x_{1}(t) = y(t)\) and \(x_{2}(t) = \dot{y}(t)\) qualify as state variables of the system at time \(t\).

(b) Compute the transfer function of the system, and use it to describe what may be special about the case \(c = 2\).

Exercise 7.3

The input \(u(t)\) and output \(y(t)\) of a system are related by the equation

\[\frac{d y(t)}{d t}+a_{0}(t) y(t)=b_{0}(t) u(t)+b_{1}(t) \frac{d u(t)}{d t}\nonumber\]

Find a linear, time varying state-space representation of this system.

Exercise 7.4

Given the periodically varying system \(x(k + 1) = A(k)x(k) + B(k)u(k)\) of period \(N\), with \(A(k +N) = A(k)\) and \(B(k +N) = B(k)\), define the sampled state \(z[k]\) and the associated extended input vector \(v[k]\) by

\[z[k]=x(k N), \quad v[k]=\left(\begin{array}{c}
u(k N) \\
u(k N+1) \\
\vdots \\
u(k N+N-1)
\end{array}\right)\nonumber\]

Now show that \(z[k + 1] = F z[k] + Gv[k]\) for constant matrices \(F\) and \(G\) (i.e. matrices independent of \(k\)) by determining \(F\) and \(G\) explicitly.

Exercise 7.5

Let the state space representations of two given systems be

\[x_{i}(k+1)=A_{i} x_{i}(k)+B_{i} u_{i}(k), \quad y_{i}(k)=C_{i} x_{i}(k), \quad i=1,2\nonumber\]

Determine a state-space representation in the form

\[\begin{aligned}
x(k+1) &=A x(k)+B u(k) \\
y(k) &=C x(k)
\end{aligned}\nonumber\]

for the new system obtained when systems 1 and 2 are interconnected (a) in series, (b) in parallel, and in a feedback loop. Assume the size of the inputs and outputs of the two systems are consistent for each of the above configuration to make sense.

Exercise 7.6

Consider a pendulum comprising a mass \(m\) at the end of a light but rigid rod of length \(r\). The angle of the pendulum from its equilibrium position is denoted by \(\theta\). Suppose a torque \(u(t)\) can be applied about the axis of support of the pendulum (e.g. suppose the pendulum is attached to the axis of an electric motor, with the current through the motor being converted to torque). A simple model for this system takes the form

\[m r^{2} \ddot{\theta}(t)+f \dot{\theta}(t)+m g r \sin \theta(t)=u\tag{t}\]

where the term \(f \dot{\theta}\) represents a frictional torque, with \(f\) being a positive coefficient, and \(g\) is the acceleration due to gravity.

(a) Find a state-space representation for this model. Is your state-space model linear? time invariant?

(b) What nominal input \(u_{o}(t)\) corresponds to the nominal motion \(\theta_{o}(t)=\Omega t\) for all \(t\), where \(\Omega\) is some fixed constant?

(c) Linearize your state-space model in (a) around the nominal solution in (b). Is the resulting model linear? Is it time invariant or periodically varying?

Exercise 7.7

Consider the horizontal motion of a particle of unit mass sliding under the influence of gravity on a frictionless wire. It can be shown that, if the wire is bent so that its height \(h\) is given by \(h(x) = V_{\alpha}(x)\), then a state-space model for the motion is given by

\[\begin{array}{l}
\dot{x}=z \\
\dot{z}=-\frac{d}{d x} V_{\alpha}(x)
\end{array}\nonumber\]

Suppose \(V_{\alpha}(x)=x^{4}-\alpha x^{2}\).

(a) Verify that the above model has \((z, x) = (0, 0)\) as equilibrium point for any \(\alpha\) in the interval \(-1 \leq \alpha \leq 1\), and it also has \((z, x)=(0, \pm \sqrt{\frac{\alpha}{2}})\) as equilibrium points when \(\alpha\) is in the interval \(0 < \alpha \leq 1\).

(b) Derive the linearized system at each of these equilibrium points.