Reference tracking

Overview

The controllers we have designed so far make the state converge to zero — that is, they make

\[x(t) \rightarrow 0 \quad\text{as}\quad t \rightarrow \infty.\]

Suppose we want the state to converge to something else — that is, to make

\[x(t) \rightarrow x_\text{des} \quad\text{as}\quad t \rightarrow \infty.\]

We will see that, under certain conditions, this is easy to do.

Reference tracking with full state feedback

Consider the dynamic model

\[\dot{m} = f(m, n)\]

where $m \in \mathbb{R}^\ell$. Suppose we linearize this model about some equilibrium point $(m_e, n_e)$ to produce the state-space model

\[\dot{x} = Ax + Bu\]

where

\[x = m - m_e \qquad\qquad u = n - n_e.\]

Suppose we design linear state feedback

\[u = -Kx\]

that would make the closed-loop system

\[\dot{x} = (A - BK) x\]

asymptotically stable — that is, that would make

\[x(t) \rightarrow 0 \qquad\text{as}\qquad t \rightarrow 0.\]

Denote the standard basis for $\mathbb{R}^\ell$ by

\[e_1 = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \qquad e_2 = \begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix} \qquad \dotsm \qquad e_\ell = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix}.\]

Suppose there is some index $i \in \{ 1, \dotsc, \ell\}$ that satisfies

\[\tag{9} f(m + e_i r, n) = f(m, n) \qquad\text{for all}\qquad r \in \mathbb{R}.\]

That is, suppose the function $f$ is constant in — or, does not vary with — the $i$’th element of $m$. Then, the following three things are true:

Invariance of equilibrium point. Since

\[\begin{align*} f(m_e + e_i r, n_e) &= f(m_e, n_e) && \qquad\text{because of (9)} \\ &= 0 && \qquad\text{because ($m_e, n_e$) is an equilibrium point}, \end{align*}\]

then $(m_e + e_i r, n_e)$ is also an equilibrium point for any $r \in \mathbb{R}$.

Invariance of error in approximation of the dynamic model. The linear model

\[\dot{x} = Ax + Bu\]

is an approximation to the nonlinear model

\[\dot{m} = f(m, n).\]

The amount of error in this approximation is

\[\begin{align*} e(m, n) &= f(m, n) - (Ax + Bu) \\ &= f(m, n) - (A (m - m_e) + B (n - n_e)). \end{align*}\]

We will show that this approximation error is constant in — or, does not vary with — the $i$’th element of $m$. Before we do so, we will prove that

\[\tag{10} A e_i r = 0 \text{ for any } r \in \mathbb{R}.\]

First, we show that the $i$’th column of $A$ is zero:

\[\begin{align*} \frac{\partial f}{\partial m_i} \Biggr\rvert_{(m_e, n_e)} &= \lim_{h \rightarrow 0} \frac{f(m_e + e_i h, n_e) - f(m_e, n_e)}{h} \\ &= \lim_{h \rightarrow 0} \frac{f(m_e, n_e) - f(m_e, n_e)}{h} \\ &= \lim_{h \rightarrow 0} \frac{0}{h} \\ &= 0. \end{align*}\]

Next, denote the columns of $A$ by

\[A = \begin{bmatrix} A_1 & A_2 & \dotsm & A_\ell \end{bmatrix}.\]

Then, we compute

\[A e_i r = \left( \sum_{j = 1}^{\ell} A_j e_{ij} \right) r = A_i r = 0.\]

Now, for the approximation error:

\[\begin{align*} e(m + e_i r, n) &= f(m + e_i r, n) - (A (m + e_i r - m_e) + B (n - n_e)) \\ &= f(m, n) - (A (m + e_i r - m_e) + B (n - n_e)) &&\qquad\text{because of (9)}\\ &= f(m, n) - (A (m + m_e) + B (n - n_e)) - A e_i r \\ &= f(m, n) - (A (m + m_e) + B (n - n_e)) &&\qquad\text{because of (10)} \\ &= e(m, n). \end{align*}\]

What this means is that our state-space model is just as accurate near $(m_e + e_i r, n_e)$ as it is near the equilibrium point $(m_e, n_e)$.

Invariance of control. Suppose we implement linear state feedback with reference tracking:

\[u = -K(x - x_\text{des})\]

where

\[x_\text{des} = e_i r\]

for any $r\in\mathbb{R}$. Let’s assume (for now) that $r$ is constant, and so $x_\text{des}$ is also constant. What will $x(t)$ converge to in this case? Let’s find out. First, we define the error

\[z = x - x_\text{des}\]

and note that

\[u = - K (x - x_\text{des}) = - K z.\]

Second, we derive an expression for the closed-loop system in terms of this error:

\[\begin{align*} \dot{z} &= \frac{d}{dt} \left(x - x_\text{des}\right) \\ &= \dot{x} - 0 \\ &= Ax + Bu \\ &= A(z + x_\text{des}) + B(-Kz) \\ &= (A - BK) z + A x_\text{des} \\ &= (A - BK) z + A e_i r \\ &= (A - BK) z && \qquad\text{because of (10)}. \end{align*}\]

This means that

\[z(t) \rightarrow 0 \qquad\text{as}\qquad t \rightarrow \infty\]

or equivalently that

\[x(t) \rightarrow x_\text{des} \qquad\text{as}\qquad t \rightarrow \infty\]

so long as all eigenvalues of $A - BK$ have negative real part — exactly the same conditions under which the closed-loop system without reference tracking would have been asymptotically stable.

Result: Reference tracking with full state feedback

Consider a system

\[\dot{m} = f(m, n)\]

that satisfies

\[f(m + e_i r, n) = f(m, n) \quad\text{for any}\quad r \in \mathbb{R}.\]

Linearize this system about an equilibrium point $(m_e, n_e)$ to produce the state-space model

\[\dot{x} = Ax + Bu\]

where

\[x = m - m_e \qquad\text{and}\qquad u = n - n_e.\]

Apply linear state feedback with reference tracking as

\[u = - K (x - x_\text{des})\]

where

\[x_\text{des} = e_i r\]

for any $r \in \mathbb{R}$. Then,

\[x(t) \rightarrow x_\text{des} \qquad\text{as}\qquad t \rightarrow \infty\]

if and only if all eigenvalues of $A - BK$ have negative real part.

Reference tracking with partial state feedback

Consider the system

\[\begin{align*} \dot{m} &= f(m, n) && \qquad\qquad\text{dynamic model} \\ o &= g(m, n) && \qquad\qquad\text{sensor model} \\ \end{align*}\]

where $m \in \mathbb{R}^\ell$. Suppose we linearize this model about some equilibrium point $(m_e, n_e)$ to produce the state-space model

\[\begin{align*} \dot{x} &= Ax+Bu \\ y &= Cx + Du \\ \end{align*}\]

where

\[x = m - m_e \qquad\qquad u = n - n_e \qquad\qquad y = o - g(m_e, n_e).\]

Suppose we design a controller

\[u = -K\widehat{x}\]

and observer

\[\widehat{x} = A\widehat{x} + Bu - L(C\widehat{x} - y)\]

that would make the closed-loop system

\[\begin{bmatrix} \dot{x} \\ \dot{x}_\text{err} \end{bmatrix} = \begin{bmatrix} A - BK & -BK \\ 0 & A - LC \end{bmatrix} \begin{bmatrix} x \\ x_\text{err} \end{bmatrix}\]

asymptotically stable — that is, that would make

\[x(t) \rightarrow 0 \qquad\text{and}\qquad x_\text{err}(t) \rightarrow 0 \qquad\text{as}\qquad t \rightarrow 0\]

where

\[x_\text{err} = \widehat{x} - x.\]

Suppose, as for reference tracking with full state feedback, that there is some index $i \in \{ 1, \dotsc, \ell\}$ for which

\[f(m + e_i r, n) = f(m, n) \qquad\text{for all}\qquad r \in \mathbb{R}.\]

This implies invariance of equilibrium point and invariance of error in approximation of the dynamic model, just like before. Suppose it is also true that, for some constant vector $g_0$, the sensor model satisfies

\[\tag{11} g(m + e_i r, n) = g_0 r + g(m, n) \qquad\text{for all}\qquad r \in \mathbb{R}.\]

Then, the following two more things are true:

Invariance of error in approximation of the sensor model. The linear model

\[y = Cx + Du\]

is an approximation to the nonlinear model

\[o = g(m, n).\]

The amount of error in this approximation is

\[\begin{align*} e(m, n) &= g(m, n) - (g(m_e, n_e) + Cx + Du) \\ &= g(m, n) - (g(m_e, n_e) + C (m - m_e) + D (n - n_e)). \end{align*}\]

We will show that this approximation error is constant in — or, does not vary with — the $i$’th element of $m$. First, we show that the $i$’th column of $C$ is $g_0$:

\[\begin{align*} \frac{\partial g}{\partial m_i} \Biggr\rvert_{(m_e, n_e)} &= \lim_{h \rightarrow 0} \frac{g(m_e + e_i h, n_e) - g(m_e, n_e)}{h} \\ &= \lim_{h \rightarrow 0} \frac{g_0 h + g(m_e, n_e) - g(m_e, n_e)}{h} && \qquad\text{because of (11)}\\ &= \lim_{h \rightarrow 0} \frac{g_0 h}{h} \\ &= g_0. \end{align*}\]

Next, denote the columns of $C$ by

\[C = \begin{bmatrix} C_1 & C_2 & \dotsm & C_\ell \end{bmatrix}.\]

Then, we compute

\[\tag{12} C e_i r = \left( \sum_{j = 1}^{\ell} C_j e_{ij} \right) r = C_i r = g_0 r.\]

Now, for the approximation error:

\[\begin{align*} e(m + e_i r, n) &= g(m + e_i r, n) - (g(m_e, n_e) + C (m + e_i r - m_e) + D (n - n_e)) \\ &= g_0 r + g(m, n) - (g(m_e, n_e) + C (m + e_i r - m_e) + D (n - n_e)) &&\text{from (11)}\\ &= g_0 r + g(m, n) - (g(m_e, n_e) + C (m - m_e) + D (n - n_e)) - C e_i r \\ &= g_0 r + g(m, n) - (g(m_e, n_e) + C (m - m_e) + D (n - n_e)) - g_0 r &&\text{from (12)}\\ &= g(m, n) - (g(m_e, n_e) + C (m - m_e) + D (n - n_e)) \\ &= e(m, n). \end{align*}\]

What this means is that our state-space model is just as accurate near $(m_e + e_i r, n_e)$ as it is near the equilibrium point $(m_e, n_e)$.

Invariance of control. Suppose we implement linear state feedback with reference tracking:

\[u = -K(\widehat{x} - x_\text{des})\]

where

\[x_\text{des} = e_i r\]

for any $r\in\mathbb{R}$. Let’s assume (for now) that $r$ is constant, and so $x_\text{des}$ is also constant. What will $x(t)$ converge to in this case? Let’s find out. First, we define the state error

\[z = x - x_\text{des}\]

and the state estimate error

\[x_\text{err} = \widehat{x} - x.\]

Second, we derive an expression for the closed-loop system in terms of these errors. Let’s start with the state error:

\[\begin{align*} \dot{z} &= \frac{d}{dt} \left(x - x_\text{des}\right) \\ &= \dot{x} - 0 \\ &= Ax + Bu \\ &= A(z + x_\text{des}) + B(-K(\widehat{x} - x)) \\ &= A(z + x_\text{des}) - BK(x_\text{err} + z) \\ &= (A - BK) z - BK x_\text{err} + A x_\text{des} \\ &= (A - BK) z - BK x_\text{err} + A e_i r \\ &= (A - BK) z - BK x_\text{err} && \qquad\text{because of (10)}. \end{align*}\]

Now, for the state estimate error:

\[\begin{align*} \dot{x}_\text{err} &= \dot{\widehat{x}} - \dot{x} \\ &= \left( A \widehat{x} + Bu - L(C\widehat{x} - y) \right) - \left(Ax + Bu\right) \\ &= A(\widehat{x} - x) - LC (\widehat{x} - x) \\ &= (A - LC) x_\text{err}. \end{align*}\]

Putting these together, we have

\[\begin{bmatrix} \dot{z} \\ \dot{x}_\text{err} \end{bmatrix} = \begin{bmatrix} A - BK & -BK \\ 0 & A - LC \end{bmatrix} \begin{bmatrix} z \\ x_\text{err} \end{bmatrix}.\]

This means that

\[z(t) \rightarrow 0 \qquad\text{and}\qquad x_\text{err}(t) \rightarrow 0 \qquad\text{as}\qquad t \rightarrow \infty\]

or equivalently that

\[x(t) \rightarrow x_\text{des} \qquad \widehat{x}(t) \rightarrow x(t) \qquad\text{as}\qquad t \rightarrow \infty\]

so long as all eigenvalues of $A - BK$ and all eigenvalues of $A - LC$ have negative real part — exactly the same conditions under which the closed-loop system without reference tracking would have been asymptotically stable.

Result: Reference tracking with partial state feedback

Consider a system

\[\begin{align*} \dot{m} &= f(m, n) && \qquad\qquad\text{dynamic model} \\ o &= g(m, n) && \qquad\qquad\text{sensor model} \\ \end{align*}\]

that satisfies

\[f(m + e_i r, n) = f(m, n) \quad\text{for any}\quad r \in \mathbb{R}\]

and

\[g(m + e_i r, n) = g_0 r + g(m, n) \qquad\text{for all}\qquad r \in \mathbb{R}\]

for some constant vector $g_0$. Linearize this system about some equilibrium point $(m_e, n_e)$ to produce the state-space model

\[\begin{align*} \dot{x} &= Ax+Bu \\ y &= Cx + Du \\ \end{align*}\]

where

\[x = m - m_e \qquad\qquad u = n - n_e \qquad\qquad y = o - g(m_e, n_e).\]

Apply the observer

\[\widehat{x} = A\widehat{x} + Bu - L(C\widehat{x} - y)\]

and the controller (with reference tracking)

\[u = -K(\widehat{x} - x_\text{des})\]

where

\[x_\text{des} = e_i r\]

for any $r \in \mathbb{R}$. Then,

\[x(t) \rightarrow x_\text{des} \qquad \widehat{x}(t) \rightarrow x(t) \qquad\text{as}\qquad t \rightarrow \infty\]

if and only if all eigenvalues of $A - BK$ and all eigenvalues of $A - LC$ have negative real part.

Tracking more than one element of the state

Our discussion of reference tracking with full state feedback and with partial state feedback has assumed that we want to track desired values of exactly one element $m_i$ of the nonlinear state $m$. All of this generalizes immediately to the case where we want to track desired values of more than one element of $m$.

In particular, suppose there are two indices $i, j \in \{1, \dotsc, \ell\}$ that satisfy

\[f(m + e_i r_i + e_j r_j, n) = f(m, n) \quad\text{for any}\quad r_i, r_j \in \mathbb{R}\]

and, in the case of partial state feedback, that also satisfy

\[g(m + e_i r_i + e_j r_j, n) = g_{0i} r_i + g_{0j} r_j + g(m, n) \quad\text{for any}\quad r_i, r_j \in \mathbb{R}\]

for constant vectors $g_{0i}$ and $g_{0j}$. Then, choosing

\[x_\text{des} = e_ir_i + e_jr_j\]

would produce the same results that were derived previously.

Choosing the desired state to keep errors small

Our proof that tracking “works” relies largely on having shown that our state-space model is just as accurate near $(m_e + e_i r, n_e)$ as it is near the equilibrium point $(m_e, n_e)$. Equivalently, it relies on having shown that this model is just as accurate near $x = x_\text{des}$ as it is near $x = 0$.

Despite this fact, it is still important to keep the state error

\[x - x_\text{des}\]

small. The reason is that the input is proportional to the error — for full state feedback as

\[u = - K (x - x_\text{des})\]

and for partial state feedback as

\[u = - K (\widehat{x} - x_\text{des}).\]

So, if error is large, the input may exceed bounds (e.g., limits on actuator torque). Since our state-space model does not include these bounds, it may be inaccurate when inputs are large.

As a consequence, it is important in practice to choose $x_\text{des}$ so that the state error

\[x - x_\text{des}\]

remains small. Here is one common way to do this, both in the case of full state feedback and partial state feedback.

Result: Choosing the desired state in the case of full state feedback

Suppose $x_\text{goal}$ is the state you actually want to achieve. Suppose $e_\text{max} > 0$ is an upper bound on the state error that you are willing to tolerate. Then, choose

\[x_\text{des} = \begin{cases} x + e_\text{max} \left(\dfrac{x_\text{goal} - x}{\|x_\text{goal} - x\|} \right) & \text{if } \| x_\text{goal} - x \| > e_\text{max}, \\ x_\text{goal} & \text{otherwise.} \end{cases}\]

Result: Choosing the desired state in the case of partial state feedback

Suppose $x_\text{goal}$ is the state you actually want to achieve. Suppose $e_\text{max} > 0$ is an upper bound on the state error that you are willing to tolerate. Then, choose

\[x_\text{des} = \begin{cases} \widehat{x} + e_\text{max} \left(\dfrac{x_\text{goal} - \widehat{x}}{\|x_\text{goal} - \widehat{x}\|} \right) & \text{if } \| x_\text{goal} - \widehat{x} \| > e_\text{max}, \\ x_\text{goal} & \text{otherwise.} \end{cases}\]