Chapter 1 Introduction

We consider input-state-output systems with a state \(x:[0,\infty )\to \mR ^n\), external input \(w:[0,\infty )\to \mR ^{m_1}\), control input \(u:[0,\infty )\to \mR ^{m_2}\), performance output \(z:[0,\infty )\to \mR ^{p_1}\) and measured output \(y:[0,\infty )\to \mR ^{p_2}\) described by

\begin{equation} \label {eq:xzy} \dot {x}=Ax+B_1w+B_2u,\qquad z=C_1x+D_{11}w+D_{12}u,\qquad y=C_2x+D_{21}w+D_{22}u, \end{equation}

with the initial condition \(x(0)=x^0\) where \(x^0\in \mR ^n\) and

\begin{gather} \label {eq:ABCDmatrices} A\in \mR ^{n\times n},~ B_1\in \mR ^{n\times m_1},~ B_2\in \mR ^{n\times m_2},\notag \\ C_1\in \mR ^{p_1\times n},~ D_{11}\in \mR ^{p_1\times m_1},~ D_{12}\in \mR ^{p_1\times m_2},~ C_2\in \mR ^{p_2\times n},~ D_{21}\in \mR ^{p_2\times m_1}, D_{22}\in \mR ^{p_2\times m_2}. \end{gather}

The objective is to find some function \(f\) so that if we choose \(u=f(y)\) (i.e. the control is a function of the measurement), then \(z\) is small whatever \(w\) is (at least for \(w\) within a certain class of functions). Of course, we have to make precise what it means for \(z\) to be small and which \(w\) we consider. Different precise formulations are relevant for different applications. We note that the function \(f\) is determined by the model (i.e. the matrices in (1)) and that \(y\) in \(u=f(y)\) will actually be the physical measurement (rather than the output of the model). Therefore, we combine the model and data.

We will study various applications which can be described by the above mathematical framework. We consider two extensive case studies throughout most of this module: a tape drive (which is used for storage of data) and a suspension system for a car. We further (to illustrate specific aspects) consider the diagnosis of lung diseases, the way that the body provides energy for muscles (which is for example crucial for the mathematical analysis is sports such as running, swimming and cycling), how microphones work and how we see colors.

1.1 Case study: control of a tape drive

Most data in the world is not stored on hard drives, but on tape drives. See Figure 1.1 for a photo of a tape drive. The tape drive contains two reels which rotate due to currents being applied to motors. This causes the tape to be transported from one reel (the supply reel) to the other reel (the take-up reel). In between the reels is the read/write head. The objectives are for the velocity at the read/write head to be some given constant and for the tension in the tape to be some given constant (both are needed for the read/write operation to be performed well).

We initially model this situation assuming that the reels (including the tape wrapped around it) is perfectly circular. This however is not strictly true. To account for this, we include a disturbance. Since we have a rotational system, it is natural to assume that this disturbance is periodic with a known period.

The rationale here is similar to what is done in statistical modeling where there usually is a model “plus noise”. Here however we assume that the additional term is deterministic and instead of assuming some statistical properties of this term (such as it being Gaussian with known mean and variance), we assume that we know that it is periodic with a known period.

A simple model for the tape drive is:

\begin{align*} M_1\dot {v}_1+d_1v_1-T&=u_1,\\ M_2\dot {v}_2+d_2v_2+T&=u_2,\\ \dot {T}+k(v_1-v_2)&=kv_e. \end{align*} Here \(v_1\) is the linear velocity at the supply reel (and the head), \(v_2\) is the linear velocity at the take-up reel, \(T\) is the tension in the tape, \(u_1\) and \(u_2\) are the controls and \(v_e\) is the periodic disturbance (with frequency \(\omega _e\)). The parameters are \(M_1,M_2,d_1,d_2,k>0\). We denote the desired velocity at the read/write head by \(r_v\) and the desired tension in the tape by \(r_T\) (here the letter \(r\) standard for “reference”). We assume that we can measure both velocities and the tension and that we know the desired velocity \(r_v\) and tension \(r_T\).

If we need to consider specific parameter values, then we make the choice

\[ M_1=0.25,\quad M_2=0.15,\quad d_1=1,\quad d_2=1,\quad k=170,\quad r_v=5,\quad r_T=0.3,\quad \omega _e=650. \]

We can write this model in the standard state space form (1.1) with

\[ x:=\bbm {v_1\\v_2\\T},\qquad w=\bbm {r_v\\r_T\\v_e},\qquad u=\bbm {u_1\\u_2},\qquad z=\bbm {v_1-r_v\\T-r_T},\qquad y=\bbm {v_1\\v_2\\T\\r_v\\r_T}, \]

(i.e. \(z_1\) is the difference between the velocity at the head and its reference and \(z_2\) is the difference between the tension in the tape and its reference) and

\begin{gather*} A=\bbm { -\frac {d_1}{M_1}&0&\frac {1}{M_1}\\ 0&-\frac {d_2}{M_2}&-\frac {1}{M_2}\\ -k&k&0 },\qquad B_1=\bbm {0&0&0\\0&0&0\\0&0&k},\qquad B_2=\bbm {\frac {1}{M_1}&0\\0&\frac {1}{M_2}\\0&0}, \\ C_1=\bbm {1&0&0\\0&0&1},\qquad D_{11}=\bbm {-1&0&0\\0&-1&0}, \\ C_2=\bbm { 1&0&0\\ 0&1&0\\ 0&0&1\\ 0&0&0\\ 0&0&0 },\qquad D_{21}=\bbm { 0&0&0\\ 0&0&0\\ 0&0&0\\ 1&0&0\\ 0&1&0 }. \end{gather*}

The exosystem

We can encode that we know that \(r_v\) and \(r_T\) are constant and that \(v_e\) is periodic with frequency \(\omega _e\) through differential equations:

\[ \dot {r}_v=0,\qquad \dot {r}_T=0,\qquad \ddot {v}_e+\omega _e^2v_e=0, \]

(where we assume that \(v_e\) is sinusoidal with frequency \(\omega _e\) rather than a general periodic function). We can re-write this in standard form as

\[ \dot {x}_e=A_ex_e,\qquad w=C_ex_e, \]

where

\[ x_e=\bbm {r_v\\r_T\\v_e\\\dot {v}_e},\qquad A_e=\bbm { 0&0&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&-\omega _e^2&0 },\qquad C_e=\bbm { 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0 }. \]

Such a system which generates the references and disturbances is called an exosystem. In general we have

\[ A_e\in \mR ^{n_e\times n_e},\qquad C_e\in \mR ^{m_1\times n_e}. \]

Writing the references and disturbances in terms of an exosystem allows us to reduce our problem to linear algebra with the matrices \(A_e\) and \(C_e\).

Unmodelled dynamics

In reality, the \(y\) which is used in \(u=f(y)\) to determine the control will be an actual physical measurement rather than a computed output of our model. This has two implications. One is that there will be measurement noise. The other is that even if we perfectly measured our physical system, the actual physical system will not be perfectly described by our simple model. To take both of these aspects into account, we instead have an additional external input \(\widetilde {w}\) and

\[ \wt =\bbm {\wt _1\\\wt _2\\\wt _3},\qquad y=\bbm {v_1+\delta _1\wt _1\\v_2+\delta _2\wt _2\\T+\delta _3\wt _3\\r_v\\r_T}, \]

where \(\wt _1\) models measurement noise and unmodelled dynamics relevant to \(v_1\), and similarly \(\wt _2\) for \(v_2\) and \(\wt _3\) for \(T\) and \(\delta _1,\delta _2,\delta _3>0\). Instead of assuming that \(v_e\) is periodic with a known period (as we did above), we can model it as an arbitrary disturbance, in that case we would include \(v_e\) in \(\wt \) rather than in \(w\).

We then more generally have a system of the following form. We now have two external inputs, one being \(w:[0,\infty )\to \mR ^{m_1}\) of which we have some knowledge, namely that it is generated by an exosystem

\[ \dot {x}_e=A_ex_e,\qquad w=C_ex_e, \]

with

\[ A_e\in \mR ^{n_e\times n_e},\qquad C_e\in \mR ^{m_1\times n_e}, \]

and \(\wt :[0,\infty )\to \mR ^{\widetilde {m}_1}\) of which we assume no such knowledge. The system is given by

\[ \dot {x}=Ax+B_1w+\widetilde {B}_1\wt +B_2u,\quad z=C_1x+D_{11}w+\widetilde {D}_{11}\wt +D_{12}u,\quad y=C_2x+D_{21}w+\widetilde {D}_{21}\wt +D_{22}u, \]

where additionally to the matrices in (1.1) we have

\[ \widetilde {B}_1\in \mR ^{n\times \widetilde {m}_1},~\widetilde {D}_{11}\in \mR ^{p_1\times \widetilde {m}_1},~\widetilde {D}_{21}\in \mR ^{p_2\times \widetilde {m}_1}. \]

For our tape drive model (assuming that we model \(v_e\) as periodic, so as part of \(w\)) we have

\[ \widetilde {B}_1=\bbm {0&0&0\\0&0&0\\0&0&0},\qquad \widetilde {D}_{11}=\bbm {0&0&0\\0&0&0},\qquad \widetilde {D}_{21}= \bbm { \delta _1&0&0\\ 0&\delta _2&0\\ 0&0&\delta _3\\ 0&0&0\\ 0&0&0 }, \]

with the other matrices as earlier.

Control cost

If our only objective is to have \(v_1-r_v\) small and \(T-r_T\) small, then the controls \(u_1\) and \(u_2\) will turn out to be very large (in fact, in the “optimal” case they will be infinite). This is clearly not physically feasible. Hence we somehow have to take the physical constraint into account that the control input \(u\) cannot be too large. One way of doing this is by defining

\[ z=\bbm {v_1-r_v\\T-r_T\\\varepsilon _1 u_1\\\varepsilon _2 u_2}, \]

where \(\varepsilon _1,\varepsilon _2>0\) (and typically “small”). The objective of making \(z\) small then means that \(v_1-r_v\), \(T-r_T\), \(\varepsilon _1 u_1\) and \(\varepsilon _2 u_2\) should be small. Since \(\varepsilon _1,\varepsilon _2\) are small, the latter two requirements really mean that \(u_1\) and \(u_2\) should not be too large. This changes the above matrices to

\[ C_1=\bbm {1&0&0\\0&0&1\\0&0&0\\0&0&0},\qquad D_{11}=\bbm {-1&0&0\\0&-1&0},\qquad D_{12}=\bbm {0&0\\0&0\\\varepsilon _1&0\\0&\varepsilon _2}, \]

and we would need to add two zero rows to \(\widetilde {D}_{11}\).

1.2 Case study: a suspension system

We consider what in the automotive industry is called a quarter-car model. This is used as a first step in designing suspension systems for cars. The quarter-car model considers only one wheel (and therefore ignores the motion of the car due to coupling of the wheels). Therefore, we might also think of designing a syspension system for a unicycle (Figure 1.2).

The vertical position, velocity and acceleration of the car are considered (since this is relevant for the suspension system). The road profile (curbs, speed bumps, potholes, general roughness of the road) is an external input. This road profile is given as a vertical position as a function of horizontal position. If we assume the car moves at constant horizontal velocity, then we can view the road profile as a function of time instead. In our modelling we will consider the derivative of this road profile with respect to time, the road velocity, as the (external) input.

We model the tyre as a spring and we include a mass for the part of the car below the suspension system (this includes the tyre, the wheel and so on). In the automotive industry these are called the unsprung spring and unsprung mass. The main part of the car (the part above the suspension system) is modelled as a mass, called the sprung mass.

The suspension system is between the two masses. We have the following differential equations:

\begin{gather*} m_s\dot {v}_s+F_u=0,\qquad m_{us}\dot {v}_{us}+F_{us}-F_u=0,\\ \dot {q}=v_{us}-v_s,\qquad \dot {F}_{us}=k_{us}(v_{us}-v_e). \end{gather*} Here \(v_s\) is the velocity of the sprung mass, \(v_{us}\) is the velocity of the unsprung mass, \(F_{us}\) is the force across the unsprung spring (i.e. the tyre) and \(q\) is the distance between the two masses plus a constant (we normalize that constant so that \(q=0\) gives the “desired” distance between the masses). The constants are \(m_s>0\) (the sprung mass), \(m_{us}>0\) (the unsprung mass) and \(k_{us}>0\) (the stiffness of the unsprung spring, i.e. the tyre). The external input \(v_e\) is the road velocity (as described above) and \(F_u\) is the force applied by the suspension system (i.e. \(F_u\) is what we have to design).

We can write this model in the standard state space form (1.1) with

\[ x:=\bbm {F_{us}\\v_{us}\\q\\v_s},\qquad w:=v_e,\qquad u:=F_u, \]

and

\[ A=\bbm {0&k_{us}&0&0\\ \frac {-1}{m_{us}}&0&0&0\\ 0&1&0&-1\\ 0&0&0&0 },\qquad B_1=\bbm {-k_{us}\\0\\0\\0},\qquad B_2=\bbm {0\\\frac {1}{m_{us}}\\0\\\frac {-1}{m_s}}. \]

There are three variables relevant for the performance output:

• • The tyre deflection \(\int v_{us}-v_e\), which is a measure for handling.

• • The suspension stroke \(q\), which shouldn’t be too large in absolute value as this would damage the car (\(q\ll 0\) means that the masses get too close and might collide, \(q\gg 0\) means that the masses are too far apart so that the suspension system gets too streched).

• • The acceleration of the sprung mass, \(\dot {v}_{s}\), which is a measure for comfort.

In terms of the state variables and the control (using the differential equations) we have that the tyre deflection is \(\frac {x_1}{k_{us}}\), the suspension stroke is \(x_3\) and the acceleration of the sprung mass equals \(\frac {-u}{m_s}\). We want all three of these quantities to be small. Which one we care about most depends on the application (for a formula one car, we would mostly be interested in tyre deflection whereas for a normal car we would relatively speaking care more about the acceleration of the sprung mass). To take this into account, we introduce weights \(r_1,r_2>0\) and define the performance output by

\[ z=\bbm {r_1\frac {x_1}{k_{us}}\\r_2x_3\\\frac {-u}{m_s}}. \]

In standard form this gives

\[ C_1=\bbm {\frac {r_1}{k_{us}}&0&0&0\\0&0&r_2&0\\0&0&0&0},\qquad D_{12}=\bbm {0\\0\\\frac {-1}{m_s}}. \]

If we need to consider specific parameter values, then we make the choice

\[ m_s=250,\quad m_{us}=35,\quad k_{us}=150,\!000,\quad r_1=\frac {k_{us}}{m_s},\quad r_2=r_1\sqrt {\frac {m_{us}}{m_s}}. \]

In principle it is possible to measure the whole state, but this comes at a cost. Easy measurements are \(q\) and \(v_{us}-v_s\) and a suspension system which uses only knowledge of these two variables is easier to physically build. In either case we need to add measurement noise/unmodelled dynamics. This gives (when measuring the whole state)

\[ \qquad C_2=I,\qquad D_{21}=\bbm {0&\delta _1&0&0&0\\0&0&\delta _2&0&0\\0&0&0&\delta _3&0\\0&0&0&0&\delta _4\\},\qquad \]

or (when measuring only \(q\) and \(v_{us}-v_s\))

\[ C_2=\bbm {0&0&1&0\\0&1&0&-1}, \qquad D_{21}=\bbm {0&\delta _1&0\\0&0&\delta _2}, \]

where \(\delta _1,\delta _2,\delta _3,\delta _4>0\). In both cases, we need to write \(v_e=w_1\) and change \(B_1\) accordingly (by adding zero columns).

1.2.1 A fixed structure suspension system

Our ultimate aim is to design an “optimal” suspension system. However to understand the issues better (and to illustrate concepts in Part I of these notes), we will also consider a suspension system consisting of a spring and a damper between the two masses. This gives (here \(k_s,d>0\))

\[ u=d(v_s-v_{us})-k_sq,\qquad \text {i.e.,}~~u=d(x_4-x_2)-k_sx_3. \]

We then obtain the closed-loop system (here we ignore the measured output \(y\) and therefore just have \(w=v_e\))

\[ \dot {x}=A_\cl x+B_\cl w,\qquad z=C_\cl x, \]

with

\[ A_\cl =\bbm {0&k_{us}&0&0\\ \frac {-1}{m_{us}}&\frac {-d}{m_{us}}&\frac {-k_s}{m_{us}}&\frac {d}{m_{us}}\\ 0&1&0&-1\\ 0&\frac {d}{m_s}&\frac {k_s}{m_s}&\frac {-d}{m_s} },\qquad B_\cl =\bbm {-k_{us}\\0\\0\\0},\qquad C_\cl =\bbm {\frac {r_1}{k_{us}}&0&0&0\\ 0&0&r_2&0\\ 0&\frac {d}{m_s}&\frac {k_s}{m_s}&\frac {-d}{m_s} }. \]

This closed-loop system describes how our performance output \(z\) depends on the external input \(w\) (here the road velocity) when we choose the control input (i.e. the suspension system) \(u\) as above.

For illustrative purposes in Part I we will also consider non-optimal (but reasonable) values for \(k_s\) and \(d\); these we pick as

\[ k_s=50,\!000,\qquad d=k_s~\sqrt {\frac {m_s+m_{us}}{k_{us}}}. \]

1.3 Examples

1.3.1 First order systems

As a simple example we will often consider the following first order system:

\begin{equation} \label {eq:into:firstorder} T\dot {x}(t)+x(t)=K_1w(t)+K_2u(t), \end{equation}

where \(T,K_1,K_2>0\). The parameter \(T\) is called the time constant and the parameters \(K_1\) and \(K_2\) the steady state gains. This can be written as an input-state system with \(n=m_1=m_2=1\) and

\[ A=-\frac {1}{T},\qquad B_1=\frac {K_1}{T},\qquad B_2=\frac {K_2}{T}. \]

If the performance output is

\[ z=\bbm {x\\\varepsilon u}, \]

where \(\varepsilon >0\), then \(p_1=2\) and

\[ C_1=\bbm {1\\0},\qquad D_{11}=\bbm {0\\0},\qquad D_{12}=\bbm {0\\\varepsilon }. \]

If we consider a measured output and want to include measurement noise/unmodelled dynamics, then we should redefine \(m_1=2\), replace \(w\) in (1.3) by \(w_1\), and

\[ B_1=\bbm {\dfrac {K_1}{T}&0}, \]

and define

\[ y=x+\delta w_2,\qquad C_2=1,\qquad D_{21}=\bbm {0&\delta },\qquad D_{22}=0. \]

1.3.2 Second order systems (low pass)

Consider the second order system

\begin{equation} \label {eq:into:secondlow} T^2\ddot {q}(t)+2\zeta T\dot {q}(t)+q(t)=K_1w(t)+K_2u(t), \end{equation}

where \(T,K_1,K_2>0\) and \(\zeta \geq 0\). The parameter \(T\) is called the time constant and \(K_1\) and \(K_2\) the steady state gains. The parameter \(\zeta \) is the damping ratio. The equation can alternatively by written as

\begin{equation} \label {eq:into:secondlow:omega0} \ddot {q}(t)+2\zeta \omega _0\dot {q}(t)+\omega _0^2q(t)=K_1\omega _0^2w(t)+K_2\omega _0^2u(t), \end{equation}

where \(\omega _0=\frac {1}{T}\) is called the natural frequency. By defining

\[ n=2,\quad m_1=m_2=1,\qquad x=\bbm {q\\\dot {q}},\quad A=\bbm {0&1\\-\omega _0^2&-2\zeta \omega _0},\quad B_1=\bbm {0\\K_1\omega _0^2},\quad B_2=\bbm {0\\K_2\omega _0^2}, \]

this can be written as an input-state system.

If the performance output is

\[ z=\bbm {q\\\varepsilon u}, \]

where \(\varepsilon >0\), then \(p_1=2\) and

\[ C_1=\bbm {1&0\\0&0},\qquad D_{11}=\bbm {0\\0},\qquad D_{12}=\bbm {0\\\varepsilon }. \]

If we consider a measured output and want to include measurement noise/unmodelled dynamics, then we should redefine \(m_1=2\), replace \(w\) in (1.4) by \(w_1\), and

\[ B_1=\bbm {0&0\\K_1\omega _0^2&0}, \]

and define

\[ y=q+\delta w_2,\qquad C_2=\bbm {1&0},\qquad D_{21}=\bbm {0&\delta },\qquad D_{22}=0. \]

1.3.3 Second order systems (band pass)

If in contrast to Section 1.3.2, we consider the performance output

\[ z=\bbm {\dot {q}\\\varepsilon u}, \]

then

\[ C_1=\bbm {0&1\\0&0},\qquad D_{11}=\bbm {0\\0},\qquad D_{12}=\bbm {0\\\varepsilon }. \]

Similarly, if the measurement is \(y=\dot {q}+\delta w_2\), then we obtain

\[ C_2=\bbm {0&1},\qquad D_{21}=\bbm {0&\delta },\qquad D_{22}=0. \]

1.4 Problems

Consider a mass \(m>0\) which is attached to a platform through a spring with spring constant \(k>0\) and which is damped proportionally to its velocity with proportionality constant \(d\geq 0\). Further assume that a control force \(u\) is applied to the mass. Assume that the platform moves with an externally given velocity \(v_e\) and that an externally given force \(F_e\) is applied to the mass. With \(q\) the displacement of the mass relative to the platform and \(v\) the velocity of the mass we have

\[ \dot {q}=v-v_e,\qquad m\dot {v}+dv+kq=F_e+u. \]

• • With the state \(x=\sbm {q\\v}\) and the external input \(w=\sbm {F_e\\v_e}\), write this in the standard form

  \[ \dot {x}=Ax+B_1w+B_2u. \]

• • Consider as performance output \(z=\sbm {q\\\varepsilon u}\) where \(\varepsilon >0\). Write this in the standard form

  \[ z=C_1x+D_{11}w+D_{12}u. \]

• • Consider as measured output \(y=q+\delta w_3\) where \(\delta >0\). Indicate how the addition of \(w_3\) changes the matrices obtained above and write the measured output in the standard form

  \[ y=C_2x+D_{21}w+D_{22}u. \]

• • Consider as measured output \(y=\dot {q}+\delta w_3\) where \(\delta >0\). Write the measured output in the standard form

  \[ y=C_2x+D_{21}w+D_{22}u. \]