EES4710/ME4300 Modern Control System

National University of Singapore(Suzhou) Research Institute Prof. Ong Chong Lin Email: mpeongcj@nus.edu.sg Tel: 6516-2217 Editor: Longbin

Chapter 0 – Review of Basic Control System

Open Loop & Close Loop

Parameter Sensitivity

\[S^T_G = \frac{\Delta T}{\Delta G}\]

Where T is the transfer function (Combination of controlled system, controller and feedback), and G is the controlled system.

Advantages of close loop control

decrease effect of input noise
decrease effect of sensor noise

Laplacian Transformation

\[L[f(t)] = \int_0^\infty f(t)*e^{-st}dt\]

Transfer Function

Input $r(t): R(s)$
Output $c(t): C(s)$
System $G(s) = \frac{C(s)}{R(s)}$

Mathematical Modeling

Example1: Mass-Spring-Damper System

\[G(s) = \frac{1}{ms^2+ks+1}\]

Example2: RLC Circuit System

\[G(s) = \frac{s}{Ls^2 + Rs + \frac{1}{C}}\]

Other definitions

Proper Transfer Function: $n \geq m$ (i.e. $G(s) = Constant\vert_{t->\infty}$)

Time Response of System

Typical Signal

Step
Ramp
Parabolic

Response of First-Order System

step: $c(t) = 1 - e^{-\frac{t}{T}}$

Second-Order System

$\omega_n$: natural frequency
$\xi$: damping ratio
$0 < \xi < 1$: Underdamped
$\omega_d = \omega_n \sqrt{1-\xi^2}$: Damped natural frequency
$\xi = 1$: critically damped
$\xi > 1$: overdamped

Transient Response Definitions

Rise time: $t_r = \frac{\pi - \phi}{\omega_d}$
Peak time: $t_p = \frac{\pi}{\omega_d}$

Stability of System

$f(t)$ is bounded if there is an M s.t. $\lvert f(t) \rvert < M < \infty$

Definition: A system is Bounded-Input Bounded-Ouput(BIBO) stable if whenever the input signal is bounded, so is the output signal.

BIBO System: All poles are in the left-half s-panel.

Routh Table

\[a_0s^n + a_1s^{n-1} + ... + a_n = 0\]

$s^n$	$a_0$	$a_2$	$a_4$	$a_6$
$s^{n-1}$	$a_1$	$a_3$	$a_5$	$a_7$
$s^{n-2}$	$b_0$	$b_1$	$b_2$	$b_3$
…	…	…	…	…

\[b_0 = \frac{a_1a_2 - a_0a_3}{a_1}\] \[b_1 = \frac{a_1a_4 - a_0a_5}{a_1}\] \[...\]

Frequency Response

\[\frac{|Y(j\omega)|}{|U(j\omega)|} = |G(j\omega)|\] \[\phi = \angle G(j\omega)\]

Chapter 1 – Introduction

Problem with classical control

Transfer function has its limitations. It can not describe the internel behavior of a system, which is more complicated than external behavior.

Mathematical Description

Class of systems

SISO: Single Input and Single Output system
MIMO: Multi Input and Multi Output system

Causality

Memoryless: output $y_{t_0}$ depends only on the input applied at time $t_0$
Most systems have memory: output $y_{t_0}$ depends on $u(t)$ for $t < t_0, t = t_0, t > t_0$
Causal/Non-anticipatory system: current output depends on past and current input

States and associated definitions

State Variables: the smallest set of variables determine the state of the system
State Vector: n state variables -> n dimensions vector $x(t)$
State Space: space of state vector
State Trajectory: the path produced by the state vector in the state space with the changes of time

Lumpedness

Lumped system: finite state variables or finite state vector

Linear System

Superposition

additivity: 可加性
homogeneity: 齐次性

\[y = y_{zero \space input} + y_{zero \space state}\]

State Space Representation

\[\dot{x}(t) = A(t)x(t) + B(t)u(t) \\ y(t) = C(t)x(t) + D(t)u(t)\]

When $A(t),B(t),C(t),D(t)$ are constants, the equations describe an LTI system:

\[\dot{x}(t) = Ax(t) + Bu(t) \\ y(t) = Cx(t) + Du(t)\]

Words in RLC circuit: Inductor: 电感 Resistor: 电阻

Chapter 2 – Review of Linear Algebra

Words in Linear Algebra

Basis: 基
orthogonal: 正交
- \[\alpha^T * \beta = 0\]
orthonormal: 单位正交
Domain: 定义域
Range: 值域
Rank: 秩
- \[R(A) = r\]
Full Rank: 满秩
- \[R(A_{n \times n}) = n\]
Non-Singular: 非奇异
Determinant: 行列式
- \[det(A) = |A|\]
Minor: 余子式
- \[M_{ij}\]
Co-Minor: 代数余子式
- \[C_{ij} = (-1)^{i+j} * M_{ij}\]
Adjugate matrix: 伴随矩阵
- \[A^*\]
Transpose: 转置
- \[A^T\]
Inverse: 逆
- \[A^{-1}\]
Eigenvalues: 特征值
- \[det(\lambda I - A) = 0\]
Eigenvector: 特征向量
- \[Ax = \lambda x\]
Similarity Transformation: 相似变换（相似矩阵）
- \[\bar{A} = Q^{-1}AQ\]
Diagonal: 对角阵
- \[\Lambda\]
Caley-Hamilton Theorem
- \[del(\lambda I - A) = \lambda^n + a_1 \lambda^{n-1} + \dots + a_n = 0 \\ \implies A^n + a_1 A^{n-1} + \dots + a_n I = 0\]

Chapter 3 – State Space

Introduction

\[\dot{x} = Ax + Bu \\ y = Cx + Du\]

Solution of $\dot{x} = Ax$

\[x(t) = e^{A(t-t_0)}x(t_0)\]

Properties of $e^{At}$

\[\frac{d}{dt}e^{At} = Ae^{At} = e^{At}A\]
\[e^{At} = \mathcal {L}^{-1}[(sI - A)^{-1}]\]
\[e^{At_1}e^{At_2} = e^{A(t_1+t_2)}\]
\[e^{At}e^{-At} = I\]
\[[e^{At}]^T = e^{A^Tt}\]
\[e^{At}e^{Bt} = e^{(A+B)t} \space \text{if and only if} \space AB = BA\]

Methods of Computing $e^{At}$

\[e^{At} = \sum_{k = 0}^{\infty} \frac{A^kt^k}{k!}\]
\[e^{At} = \mathcal{L}^{-1}[(sI - A)^{-1}]\]
\[e^{At} = Qe^{\Lambda t}Q^{-1}\]

Full Solution

\[y(t) = Ce^{A(t-t_0)}x(t_0) + C\int_{t_0}^t e^{A(t-\tau)}Bu(\tau)d\tau + Du(t)\]

Non-uniqueness of representation and Invariance of Eigenvalues

\[x = Q\bar{x} \\ \implies \bar{A} = Q^{-1}AQ, \bar{B} = Q^{-1}B, \bar{C} = CQ, \bar{D} = D\]

After the transformation, eigenvalues don’t change.

State Space to Transfer Function

\[G(s) = C(sI - A)^{-1}B + D\]

Chapter 4 – Controllability and Observability

Controllability

$\text{Consider}$

\[U = [B \space AB \space \dots \space A^{n-1}B]\]

$\text{System is controllable if and only if}$

\[R(U) = n\]

Observability

$\text{Consider}$

\[O = \left ( \begin{matrix} C \\ CA \\ \vdots \\ CA^{n-1} \end{matrix} \right )\]

$\text{System is observable if and only if}$

\[R(O) = n\]

Output Controllability

$\text{Consider}$

\[P = [CB \space CAB \space \dots \space CA^{n-1}B]\]

$\text{System is observable if and only if}$

\[R(P) = m\]

Chapter 5 – Stability

BIBO Stability

\[|u(t)| \leq k_1 < \infty, \forall t \ge 0, |y(t)| \leq k_2 < \infty\]

A SISO system is stable, if and only if:

\[\int_0^\infty |g(t)|dt \leq k < \infty\]

The T.F. of a SISO system is $G(s)$. It’s stable, if and only if all the poles of $G(s)$ are in the open left-half s-plane.

Internal Stability

$x_e$ is equilibrium point, if:

\[Ax_e = 0\]

A system is internal stable, if and only if:

\[x_e = 0 (origin)\]

Lyapunov Stability

$\text{Lyapunov Function}$

\[V(x)\]

$\text{System is asymptotically stable, if}$

\[\frac{dV(x)}{dt} < 0, \forall x \neq 0\]

Usually $V(x) = x^T P x, P > 0$

Lyapunov Equation

\[A^TP + PA = -Q\]

The system is stable, if and only if:

\[\forall Q > 0, \exists P > 0\]

Usually let $Q = I$, then solve the equation. The system is stable, if P > 0.

Chapter 6 – Canonical Forms

\[G(s) = \frac{b_{n-1}s^{n-1} + \dots + b_1s + b_0}{s^n + a_{n-1}s^{n-1} + \dots + a_1s + a_0}\]

Controllable Canonical Form

If the system is controllable, it can be transformed into a controllable canonical form:

\[A = \left[ \begin{matrix} 0 & 1 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 1 \\ -a_0 & -a_1 & \dots & -a_{n-1} \end{matrix} \right], B = \left[ \begin{matrix} 0 \\ 0 \\ \vdots \\ 1 \end{matrix} \right]\] \[C = \left[ \begin{matrix} b_0 & b_1 & \dots & b_{n-1} \end{matrix} \right]\]

Let the transformation be $\bar{x} = Tx$.

\[T = \left[ \begin{matrix} q^T \\ q^TA \\ \vdots \\ q^TA^{n-1} \end{matrix} \right]\] \[q^T \space \text{is the last row of} \space U^{-1}\]

Observable Canonical Form

If the system is observable, it can be transformed into a controllable canonical form:

\[A = \left[ \begin{matrix} 0 & 0 & \dots & 0 & -a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & \dots & 1 & -a_{n-1} \end{matrix} \right], B = \left[ \begin{matrix} b_0 \\ b_1 \\ \vdots \\ b_{n-1} \end{matrix} \right]\] \[C = \left[ \begin{matrix} 0 & 0 & \dots & 1 \end{matrix} \right]\]

Let the transformation be $x = Q\bar{x}$.

\[Q = \left[ \begin{matrix} \omega & A\omega & \dots & A^{n-1}\omega \end{matrix} \right]\] \[\omega \space \text{is the last column of} \space O^{-1}\]

Realization

It is easy to find a S.S. representation with given $G(s)$.

Chapter 7 – State Feedback and Observer

State Feedback

\[u = kx + r \\ k \space \text{is known as feedback gain} \\ k = [k_n \space k_{n-1} \dots k_1]\]

close-loop eigenvalues:

\[det(sI - (A + Bk)) = 0\]

With desired characteristic equation:

\[s^n + \hat{\alpha}_1s^{n-1} + \dots + \hat{\alpha}_n = 0\]

Solve $k$ with:

\[det(sI - (A + Bk)) = s^n + \hat{\alpha}_1s^{n-1} + \dots + \hat{\alpha}_n\]

Steps of the solution (desired eigenvalues: $\hat{\lambda}_i$):

Find $det(sI - A) = s^n + \alpha_1s^{n-1} + \dots + \alpha_n$
Compute $T$ that transforms $(A,B)$ to CCF
Compute $\Pi_{i=1}^n(s - \hat{\lambda}_i) = s^n + \hat{\alpha}_1s^{n-1} + \dots + \hat{\alpha}_n$
Compute $\bar{k} = [(\alpha_n - \hat{\alpha}_n), (\alpha_{n-1} - \hat{\alpha}_{n-1}), \dots, (\alpha_1 - \hat{\alpha}_1)]$
\[k = \bar{k}T\]

Luenberger Linear Observer

\[\dot{\hat{x}} = A\hat{x} + Bu + L(y - \hat{y}) \\ e = x - \hat{x} \\ \dot{e} = (A - LC)e\]

Same as State Feedback, find the solution:

\[det(sI - (A - LC)) = \Pi_{i=0}^n(s-\hat{\lambda}_i)\]

Steps of the solution (desired eigenvalues: $\hat{\lambda}_i$):

Find $det(sI - A) = s^n + \alpha_1s^{n-1} + \dots + \alpha_n$
Compute $Q$ that transforms $(A,B)$ to OCF
Compute $\Pi_{i=1}^n(s - \hat{\lambda}_i) = s^n + \hat{\alpha}_1s^{n-1} + \dots + \hat{\alpha}_n$
Compute $\bar{L} = [(\hat{\alpha}_n - \alpha_n), (\hat{\alpha}_{n-1} - \alpha_{n-1}), \dots, (\hat{\alpha}_1 - \alpha_1)]^T$
\[L = Q\bar{L}\]

Combination of State Feedback and Observer

\[\left( \begin{matrix} \dot{x} \\ \dot{e} \end{matrix} \right) = \left( \begin{matrix} A+BK & -BK \\ 0 & A-LC \end{matrix} \right) \left( \begin{matrix} x \\ e \end{matrix} \right) + \left( \begin{matrix} B \\ 0 \end{matrix} \right) r\] \[y = \left( \begin{matrix} C & 0 \end{matrix} \right) \left( \begin{matrix} x \\ e \end{matrix} \right)\]

Computer Control I

Shannon’s Sampling Theorem

\[\omega_s > 2 \omega_c\] \[\text{Nyquist frequency} \space \omega_N = \frac{\omega_s}{2}\]

Zero-Order Hold

\[\hat{u}(t) = u(t_k), t_k \leq t < t_{k+1}\]

Second-Order Filter

\[G(s) = \frac{\omega^2}{s^2 + 2\zeta \omega s + \omega^2}\]

Computer Control II

Discrete-Time State Space

\[x(k+1) = \Phi x(k) + \Gamma u(k) \\ y(k) = C x(k) + D u(k)\]

$\text{Where}$

\[\Phi = e^{Ah}, \Gamma = \int_0^h e^{At}dtB \\ h: \text{sampling period}\]

Computation of $\Phi$ and $\Gamma$

Same as the Continuous-Time State Space One Simplification:

\[\Psi = \int_0^h e^{At}dt \\ \varPhi = I + A \Psi, \Gamma = \Psi B\]

The Inverse of Sampling

\[A = \frac{\ln(\Phi)}{h} = \ln(\Phi^{\frac{1}{k}}) \\ B = \Psi^{-1} \Gamma = (A \Psi)^{-1}A\Gamma = (\Phi - I)^{-1} A \Gamma\]

Z Transform

\[F(z) = \sum_{k=0}^\infty f(kh)z^{-k} \\ f(kh) = \frac{1}{2\pi i}\int F(z)z^{k-1}dz\]

Direct Conversion

\[H(z) = \frac{Y(z)}{U(z)} = C(zI-\Phi)^{-1} \Gamma\]

Hold-Equivalent D.T.T.F

\[H(z) = (1-z^{-1})\mathcal{Z}[\frac{G(s)}{s}]\]

Poles and Zeros

\[z_i = e^{s_ih}\]

Computer Control III

Stability

Table Method

\[a_i^{k-1} = a_i^k - \alpha_k a_{k-i}^k \\ \alpha_k = \frac{a_k^k}{a_0^k}\]

Lyapunov Second Method

\[\Phi^TP\Phi - P = -Q\]

Controllability of Discrete system

\[U_d = [\Gamma \space \Phi \Gamma \space \dots \space \Phi^{n-1}\Gamma]\]

Observability of Discrete system

\[O_d = \left( \begin{matrix} C \\ C \Phi \\ \vdots \\ C \Phi^{n-1} \end{matrix} \right)\]

Canonical Forms

Same as Continuous system.

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EES4710/ME4300 Modern Control System