ECE 488: Multivariable Control Systems

Yash Pant

Estimated study time: 55 minutes

Table of contents

Sources and References

Primary textbook — J. P. Hespanha, Linear Systems Theory, 2nd ed., Princeton University Press, 2018. Supplementary texts — S. P. Bhattacharya and L. H. Keel, Linear Multivariable Control Systems, Cambridge University Press, 2022; C.-T. Chen, Linear System Theory and Design, 4th ed., Oxford University Press, 2012. Online resources — MIT OpenCourseWare 6.241J Dynamic Systems and Control; lecture notes by F. Paganini (Univ. ORT Uruguay) on linear systems.

Chapter 0: Introduction and Review

Signals and Systems

A signal is a function of time. In continuous time, signals are maps \( u : \mathbb{R} \to \mathbb{R}^m \); in discrete time they are sequences \( u : \mathbb{Z} \to \mathbb{R}^m \). A system is a map from input signals to output signals. This course is primarily concerned with finite-dimensional, linear, time-invariant (LTI) systems with multiple inputs and outputs.

SISO vs. MIMO

A SISO (single-input single-output) system has scalar input \( u(t) \in \mathbb{R} \) and scalar output \( y(t) \in \mathbb{R} \). A MIMO (multiple-input multiple-output) system has \( m \)-dimensional input \( u(t) \in \mathbb{R}^m \) and \( p \)-dimensional output \( y(t) \in \mathbb{R}^p \). The central difficulty in MIMO control is that inputs and outputs are coupled — steering one output may disturb others.

Transfer Functions

For an LTI SISO system with impulse response \( h(t) \), the transfer function is the Laplace transform

\[ H(s) = \int_0^\infty h(t) e^{-st} \, dt. \]

For a MIMO LTI system with impulse response matrix \( H(t) \in \mathbb{R}^{p \times m} \), the transfer matrix is

\[ \mathbf{H}(s) = \int_0^\infty H(t) e^{-st} \, dt \in \mathbb{R}^{p \times m}(s), \]

where \( \mathbb{R}^{p \times m}(s) \) denotes the ring of \( p \times m \) matrices of proper rational functions in the complex variable \( s \).

Block Diagrams

Block diagrams represent interconnections of systems. In feedback:

\[ Y(s) = \mathbf{P}(s) U(s), \quad U(s) = \mathbf{C}(s)\bigl(R(s) - Y(s)\bigr), \]

giving the closed-loop transfer matrix

\[ \mathbf{T}(s) = \bigl(I + \mathbf{P}(s)\mathbf{C}(s)\bigr)^{-1} \mathbf{P}(s)\mathbf{C}(s). \]

BIBO Stability

A system is BIBO stable (bounded-input bounded-output) if every bounded input produces a bounded output. For an LTI system with impulse response \( H(t) \), BIBO stability holds if and only if \[ \int_0^\infty \|H(t)\|_1 \, dt < \infty, \]

where \( \|\cdot\|_1 \) denotes the entry-wise \( \ell^1 \) norm of the matrix. In terms of the transfer function, BIBO stability is equivalent to all poles of \( \mathbf{H}(s) \) lying in the open left-half plane \( \{\text{Re}(s) < 0\} \).

Feedback Stability

A feedback system formed by plant \( \mathbf{P}(s) \) and controller \( \mathbf{C}(s) \) is internally stable if the closed-loop map from all exogenous inputs to all internal signals is BIBO stable. Internal stability is a stronger requirement than mere closed-loop pole placement in the LHP.

Chapter 1: System Representation

General State-Space Models

The standard continuous-time state-space model is

\[ \dot{x}(t) = f(x(t), u(t), t), \quad y(t) = g(x(t), u(t), t), \]

where \( x(t) \in \mathbb{R}^n \) is the state, \( u(t) \in \mathbb{R}^m \) is the input, and \( y(t) \in \mathbb{R}^p \) is the output. For LTI systems the vector fields \( f \) and \( g \) are linear and time-independent:

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

with \( A \in \mathbb{R}^{n \times n} \), \( B \in \mathbb{R}^{n \times m} \), \( C \in \mathbb{R}^{p \times n} \), \( D \in \mathbb{R}^{p \times m} \). The quadruple \( (A, B, C, D) \) is called a realization.

Discrete-Time Models

The discrete-time counterpart is

\[ x_{k+1} = A_d x_k + B_d u_k, \quad y_k = C_d x_k + D_d u_k. \]

When obtained by zero-order hold (ZOH) sampling of a continuous-time system with period \( T \),

\[ A_d = e^{AT}, \quad B_d = \int_0^T e^{A\tau} B \, d\tau. \]

Linearization

Consider a nonlinear system \( \dot{x} = f(x, u) \). Let \( (x_e, u_e) \) be an equilibrium: \( f(x_e, u_e) = 0 \). Define perturbations \( \delta x = x - x_e \), \( \delta u = u - u_e \). Taylor expansion gives

\[ \delta\dot{x} \approx \underbrace{\frac{\partial f}{\partial x}\bigg|_{x_e,u_e}}_{A} \delta x + \underbrace{\frac{\partial f}{\partial u}\bigg|_{x_e,u_e}}_{B} \delta u. \]

The Jacobian matrices \( A \) and \( B \) define the linearized system, valid for small deviations from the equilibrium.

LTI System Characterization

An LTI system is time-invariant if shifting the input by \( \tau \) shifts the output by \( \tau \). It is linear if superposition holds: the response to \( \alpha u_1 + \beta u_2 \) equals \( \alpha y_1 + \beta y_2 \).

Key properties of an LTI system \( \dot{x} = Ax + Bu, \; y = Cx + Du \):

Eigenvalues of \( A \) determine the modes of the unforced system.
The system is asymptotically stable iff all eigenvalues of \( A \) lie in the open LHP.
The transfer matrix is \( \mathbf{H}(s) = C(sI - A)^{-1}B + D \).

Transfer Functions and Realizations

Given a proper rational transfer matrix \( \mathbf{H}(s) \), a realization \( (A, B, C, D) \) is any state-space model such that \( C(sI-A)^{-1}B + D = \mathbf{H}(s) \). Realizations are not unique; two realizations are equivalent if related by a state transformation \( T \):

\[ (\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D}) = (TAT^{-1}, TB, CT^{-1}, D). \]

A realization is minimal if no realization of lower order exists; equivalently, a realization is minimal iff it is both controllable and observable (proved later).

Solutions to Linear Systems

Matrix Exponential

The matrix exponential of \( A \in \mathbb{R}^{n \times n} \) is defined by the power series

\[ e^{At} = \sum_{k=0}^{\infty} \frac{(At)^k}{k!} = I + At + \frac{A^2 t^2}{2!} + \cdots \]

The series converges absolutely for all \( t \in \mathbb{R} \) and all \( A \). Key properties:

\( \frac{d}{dt} e^{At} = A e^{At} = e^{At} A \).
\( e^{A \cdot 0} = I \).
\( e^{A(t+s)} = e^{At} e^{As} \).
\( (e^{At})^{-1} = e^{-At} \).
If \( AB = BA \) then \( e^{(A+B)t} = e^{At} e^{Bt} \).

Computation via Jordan form. Suppose \( A = PJP^{-1} \) where \( J \) is the Jordan normal form. Then \( e^{At} = P e^{Jt} P^{-1} \). For a Jordan block \( J_k(\lambda) \) of size \( k \) with eigenvalue \( \lambda \):

\[ e^{J_k(\lambda) t} = e^{\lambda t} \begin{pmatrix} 1 & t & \frac{t^2}{2!} & \cdots & \frac{t^{k-1}}{(k-1)!} \\ 0 & 1 & t & \cdots & \frac{t^{k-2}}{(k-2)!} \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \cdots & 1 & t \\ 0 & 0 & \cdots & 0 & 1 \end{pmatrix}. \]

State-Space Solution Formula

The unique solution to \( \dot{x}(t) = Ax(t) + Bu(t) \) with initial condition \( x(0) = x_0 \) is \[ x(t) = e^{At} x_0 + \int_0^t e^{A(t-\tau)} B u(\tau) \, d\tau. \]

The output is \( y(t) = C e^{At} x_0 + \int_0^t C e^{A(t-\tau)} B u(\tau) \, d\tau + D u(t) \).

Define \( \phi(t) = e^{-At} x(t) \). Then \[ \dot{\phi}(t) = -A e^{-At} x(t) + e^{-At} \dot{x}(t) = e^{-At}(\dot{x}(t) - Ax(t)) = e^{-At} Bu(t). \]

Integrating from \( 0 \) to \( t \):

\[ \phi(t) - \phi(0) = \int_0^t e^{-A\tau} B u(\tau) \, d\tau. \]

Since \( \phi(0) = x_0 \), multiplying both sides by \( e^{At} \) gives the result.

The term \( e^{At} x_0 \) is the zero-input response (free response); the convolution integral is the zero-state response (forced response). The impulse response matrix is \( H(t) = C e^{At} B \) for \( t > 0 \) (plus \( D \delta(t) \) if \( D \neq 0 \)).

Chapter 2: Stabilization via Lyapunov Theory

Vector and Matrix Norms

A vector norm \( \|\cdot\| : \mathbb{R}^n \to \mathbb{R}_{\geq 0} \) satisfies: (i) \( \|x\| \geq 0 \) with equality iff \( x = 0 \); (ii) \( \|\alpha x\| = |\alpha| \|x\| \); (iii) triangle inequality \( \|x + y\| \leq \|x\| + \|y\| \). Common norms: \( \|x\|_1 = \sum_i |x_i| \), \( \|x\|_2 = \sqrt{x^\top x} \), \( \|x\|_\infty = \max_i |x_i| \).

The induced (operator) norm of \( A \in \mathbb{R}^{n \times n} \) with respect to a vector norm \( \|\cdot\| \) is \[ \|A\| = \sup_{x \neq 0} \frac{\|Ax\|}{\|x\|} = \sup_{\|x\|=1} \|Ax\|. \]

The induced 2-norm equals the largest singular value of \( A \): \( \|A\|_2 = \sigma_{\max}(A) \).

All norms on finite-dimensional spaces are equivalent: there exist constants \( c_1, c_2 > 0 \) such that \( c_1 \|x\|_\alpha \leq \|x\|_\beta \leq c_2 \|x\|_\alpha \) for all \( x \).

Lyapunov Stability

Consider the autonomous system \( \dot{x} = f(x) \) with equilibrium \( x_e = 0 \). The equilibrium is:

Lyapunov stable if for every \( \varepsilon > 0 \) there exists \( \delta > 0 \) such that \( \|x(0)\| < \delta \Rightarrow \|x(t)\| < \varepsilon \) for all \( t \geq 0 \).
Asymptotically stable if it is Lyapunov stable and \( \|x(t)\| \to 0 \) as \( t \to \infty \).
Globally asymptotically stable (GAS) if it is asymptotically stable for all initial conditions.

Lyapunov’s Direct Method

(Lyapunov Stability Theorem) Let \( V : \mathbb{R}^n \to \mathbb{R} \) be a continuously differentiable function with \( V(0) = 0 \) and \( V(x) > 0 \) for \( x \neq 0 \). If \[ \dot{V}(x) = \nabla V(x)^\top f(x) \leq 0 \quad \forall x, \]

then the origin is Lyapunov stable. If \( \dot{V}(x) < 0 \) for \( x \neq 0 \), the origin is asymptotically stable.

For LTI systems \( \dot{x} = Ax \), choose a quadratic Lyapunov function \( V(x) = x^\top P x \) with \( P = P^\top \succ 0 \) (positive definite). Then

\[ \dot{V}(x) = \dot{x}^\top P x + x^\top P \dot{x} = x^\top A^\top P x + x^\top P A x = x^\top (A^\top P + PA) x. \]

(Lyapunov Equation Theorem) The LTI system \( \dot{x} = Ax \) is asymptotically stable if and only if for every \( Q = Q^\top \succ 0 \), the Lyapunov equation \[ A^\top P + PA + Q = 0 \]

has a unique solution \( P = P^\top \succ 0 \).

Sufficiency: Given \( P \succ 0 \) solving the equation, \( V(x) = x^\top P x \) is a valid Lyapunov function: \[ \dot{V}(x) = x^\top(A^\top P + PA) x = -x^\top Q x \leq -\lambda_{\min}(Q)\|x\|^2 < 0 \quad \text{for } x \neq 0. \]

Hence the origin is GAS.

Necessity: If \( A \) is asymptotically stable, all eigenvalues lie in the open LHP, so \( e^{At} \to 0 \) exponentially. Define

\[ P = \int_0^\infty e^{A^\top t} Q e^{At} \, dt, \]

which converges since \( \|e^{At}\| \leq C e^{-\alpha t} \) for some \( C, \alpha > 0 \). Then \( P \succ 0 \) (since \( Q \succ 0 \)) and one verifies by differentiating that \( A^\top P + PA = -Q \).

The explicit solution is \( P = \int_0^\infty e^{A^\top \tau} Q e^{A\tau} d\tau \).

BIBO vs. Lyapunov Stability

For LTI systems, the relationship between BIBO and Lyapunov (asymptotic) stability depends on the realization:

Asymptotic stability (all eigenvalues of \( A \) in open LHP) implies BIBO stability, provided \( C \) and \( B \) are finite.
BIBO stability does NOT necessarily imply asymptotic stability of the state-space system, because there may be unstable modes that are hidden (cancelled in the transfer function).

Consider the system with \( A = 1 \), \( B = 0 \), \( C = 0 \), \( D = 1 \). The transfer function is \( H(s) = 1 \) (BIBO stable), but the unobservable/uncontrollable state \( \dot{x} = x \) is unstable. The state grows unboundedly even though the output remains bounded.

For a minimal realization (fully controllable and observable), BIBO stability and asymptotic stability are equivalent. The poles of the transfer function coincide with the eigenvalues of \( A \).

Discrete-Time Stability

For the discrete-time system \( x_{k+1} = A_d x_k \), the origin is asymptotically stable iff all eigenvalues of \( A_d \) lie strictly inside the unit disk \( \{|z| < 1\} \). The discrete-time Lyapunov equation is

\[ A_d^\top P A_d - P + Q = 0, \]

and it has a unique positive definite solution iff \( A_d \) is Schur stable (all eigenvalues inside the unit disk).

Chapter 3: Controllability and State Feedback

Controllable and Reachable Subspaces

A state \( x_1 \in \mathbb{R}^n \) is reachable from the origin in time \( t_f \) if there exists a control input \( u : [0, t_f] \to \mathbb{R}^m \) such that \( x(0) = 0 \) and \( x(t_f) = x_1 \). The set of all reachable states is the reachable subspace \( \mathcal{R} \).

A state \( x_0 \in \mathbb{R}^n \) is controllable to the origin in time \( t_f \) if there exists a control such that \( x(0) = x_0 \) and \( x(t_f) = 0 \). The controllable subspace is the set of all such \( x_0 \).

For LTI systems, the reachable and controllable subspaces are equal and time-independent (for any \( t_f > 0 \)). We refer to both simply as the controllable subspace.

Controllability Gramian

The controllability Gramian over \( [0, t_f] \) is \[ W_c(t_f) = \int_0^{t_f} e^{A\tau} B B^\top e^{A^\top \tau} \, d\tau. \]

The reachable subspace equals the column space of \( W_c(t_f) \) for any \( t_f > 0 \): \( \mathcal{R} = \text{Im}(W_c(t_f)) \). The system is completely controllable iff \( W_c(t_f) \succ 0 \) (equivalently, \( \text{rank}(W_c(t_f)) = n \)) for some (hence all) \( t_f > 0 \).

Open-Loop Minimum Energy Control

Given that \( (A,B) \) is controllable, the minimum-energy input to steer from \( x(0) = 0 \) to \( x(t_f) = x_1 \) is:

Among all inputs \( u(\cdot) \) steering the state from \( 0 \) to \( x_1 \) in time \( t_f \), the one minimizing the energy \( \int_0^{t_f} \|u(t)\|^2 \, dt \) is \[ u^*(t) = B^\top e^{A^\top(t_f - t)} W_c(t_f)^{-1} x_1, \quad t \in [0, t_f], \]

and the minimum energy is \( J^* = x_1^\top W_c(t_f)^{-1} x_1 \).

The reachability constraint is \( \int_0^{t_f} e^{A(t_f-\tau)} B u(\tau) \, d\tau = x_1 \). Define the linear operator \( \mathcal{L} : u \mapsto \int_0^{t_f} e^{A(t_f - \tau)} B u(\tau) d\tau \). Its adjoint satisfies \( (\mathcal{L}^* v)(t) = B^\top e^{A^\top(t_f-t)} v \). Using the projection theorem in Hilbert space, the minimum-norm element in the preimage \( \mathcal{L}^{-1}(x_1) \) lies in the range of \( \mathcal{L}^* \), giving \( u^* = \mathcal{L}^* v \) for some \( v \). The constraint \( \mathcal{L} u^* = x_1 \) yields \( \mathcal{L}\mathcal{L}^* v = x_1 \), i.e., \( W_c(t_f) v = x_1 \), so \( v = W_c(t_f)^{-1} x_1 \).

As the eigenvalues of \( A \) move further into the RHP (increasingly unstable modes), \( W_c^{-1} \) becomes ill-conditioned and \( J^* \) grows, meaning more energy is required to control highly unstable systems.

Controllability Matrix (Kalman Rank Test)

(Kalman Controllability Rank Condition) The pair \( (A, B) \) is controllable if and only if the controllability matrix \[ \mathcal{C} = \begin{bmatrix} B & AB & A^2 B & \cdots & A^{n-1} B \end{bmatrix} \in \mathbb{R}^{n \times nm} \]

has rank \( n \).

By the Cayley–Hamilton theorem, \( A^n = -a_{n-1}A^{n-1} - \cdots - a_0 I \) (where \( a_i \) are the coefficients of the characteristic polynomial). Hence \( e^{A\tau} = \sum_{k=0}^{n-1} \alpha_k(\tau) A^k \) for scalar functions \( \alpha_k \). Therefore \[ W_c(t_f) = \sum_{k=0}^{n-1} \sum_{j=0}^{n-1} \left(\int_0^{t_f} \alpha_k(\tau)\alpha_j(\tau) d\tau\right) A^k B B^\top (A^\top)^j. \]

It follows that \( \text{Im}(W_c) \subseteq \text{Im}(\mathcal{C}) \). For the reverse inclusion, one shows that \( \ker(W_c) = \ker(\mathcal{C}^\top) \), completing the equivalence.

PBH Controllability Test

(Popov–Belevitch–Hautus Test) The pair \( (A, B) \) is controllable if and only if for every eigenvalue \( \lambda \) of \( A \) (over \( \mathbb{C} \)), \[ \text{rank}\begin{bmatrix} \lambda I - A & B \end{bmatrix} = n. \]

This test is often more practical than the Kalman rank condition for large systems. An eigenvalue \( \lambda \) is uncontrollable iff there exists a left eigenvector \( v^\top A = \lambda v^\top \) with \( v^\top B = 0 \).

Discrete-Time Controllability

For \( x_{k+1} = A_d x_k + B_d u_k \), the controllability matrix is

\[ \mathcal{C}_d = \begin{bmatrix} B_d & A_d B_d & \cdots & A_d^{n-1} B_d \end{bmatrix}, \]

and the system is controllable iff \( \text{rank}(\mathcal{C}_d) = n \). The PBH test applies analogously.

Feedback Stabilization and Eigenvalue Assignment

(Eigenvalue Assignment) The pair \( (A, B) \) is controllable if and only if there exists a state feedback gain \( K \in \mathbb{R}^{m \times n} \) such that the closed-loop matrix \( A - BK \) has any specified set of \( n \) eigenvalues (symmetric with respect to the real axis, for real \( K \).

The control law \( u = -Kx + r \) places the closed-loop eigenvalues arbitrarily. For SISO systems, Ackermann’s formula provides the gain explicitly:

\[ K = e_n^\top \mathcal{C}^{-1} p(A), \]

where \( e_n = [0 \cdots 0 \; 1]^\top \) and \( p(\lambda) = \prod_{i=1}^n (\lambda - \lambda_i^*) \) is the desired characteristic polynomial.

Stabilizability

The pair \( (A, B) \) is stabilizable if there exists \( K \) such that \( A - BK \) is Hurwitz (all eigenvalues in the open LHP). Equivalently, by PBH: \( \text{rank}[\lambda I - A, B] = n \) for all \( \lambda \) with \( \text{Re}(\lambda) \geq 0 \).

Stabilizability is weaker than controllability: uncontrollable modes must already be asymptotically stable.

Chapter 4: Observability and Output Feedback

Unobservable Subspace

A state \( x_0 \) is unobservable if the output response to \( x(0) = x_0, u \equiv 0 \) is identically zero: \( y(t) = Ce^{At}x_0 = 0 \) for all \( t \geq 0 \). The set of all unobservable states forms a subspace \( \mathcal{N}_o \) (the unobservable subspace), which is \( A \)-invariant.

Dually, the unconstructable subspace consists of states that cannot be determined from future output measurements.

Observability Gramian

The observability Gramian over \( [0, t_f] \) is \[ W_o(t_f) = \int_0^{t_f} e^{A^\top \tau} C^\top C e^{A\tau} \, d\tau. \]

The pair \( (C, A) \) is observable if and only if \( W_o(t_f) \succ 0 \) for some (hence all) \( t_f > 0 \).

Observability Matrix (Kalman Rank Test)

The pair \( (C, A) \) is observable if and only if the observability matrix \[ \mathcal{O} = \begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^{n-1} \end{bmatrix} \in \mathbb{R}^{np \times n} \]

has rank \( n \).

PBH Observability Test

The pair \( (C, A) \) is observable if and only if for every eigenvalue \( \lambda \) of \( A \), \[ \text{rank}\begin{bmatrix} \lambda I - A \\ C \end{bmatrix} = n. \]

Duality

(Controllability–Observability Duality) The pair \( (A, B) \) is controllable if and only if the dual pair \( (A^\top, B^\top) \) is observable. Equivalently, \( (C, A) \) is observable iff \( (A^\top, C^\top) \) is controllable.

This duality means that every result about controllability has a dual result about observability, obtained by transposing matrices.

Kalman Decomposition

For a general (not necessarily controllable or observable) LTI system, there exists a coordinate transformation \( T \) such that in new coordinates the system decomposes into four canonical subsystems:

\[ \tilde{A} = T A T^{-1} = \begin{pmatrix} A_{co} & 0 & A_{13} & 0 \\ A_{21} & A_{\bar{c}o} & A_{23} & A_{24} \\ 0 & 0 & A_{c\bar{o}} & 0 \\ 0 & 0 & A_{43} & A_{\bar{c}\bar{o}} \end{pmatrix}, \]

where:

\( A_{co} \): controllable and observable subsystem (visible in the transfer function).
\( A_{\bar{c}o} \): uncontrollable but observable.
\( A_{c\bar{o}} \): controllable but unobservable.
\( A_{\bar{c}\bar{o}} \): uncontrollable and unobservable.

The transfer matrix of the system equals the transfer matrix of the \( co \) subsystem alone. Hence, the poles of the transfer matrix are the eigenvalues of \( A_{co} \), generally a subset of the eigenvalues of \( A \).

Detectability

The pair \( (C, A) \) is detectable if there exists an observer gain \( L \) such that \( A - LC \) is Hurwitz. Equivalently by PBH: \( \text{rank}[\lambda I - A^\top, C^\top] = n \) for all \( \lambda \) with \( \text{Re}(\lambda) \geq 0 \).

Detectability means that all unstable modes are observable.

State Estimation: The Luenberger Observer

Given output measurements \( y = Cx + Du \) but not direct state access, a Luenberger observer reconstructs the state:

\[ \dot{\hat{x}} = A\hat{x} + Bu + L(y - C\hat{x} - Du) = (A - LC)\hat{x} + Bu + Ly. \]

The estimation error \( e = x - \hat{x} \) satisfies

\[ \dot{e} = (A - LC) e. \]

If \( (C, A) \) is observable, the gain \( L \in \mathbb{R}^{n \times p} \) can be chosen to place the eigenvalues of \( A - LC \) anywhere (by duality with state feedback). If \( (C, A) \) is merely detectable, the unstable modes are observable and can still be estimated.

Stabilization via Output Feedback (Separation Principle)

Combining state feedback \( u = -K\hat{x} \) with the Luenberger observer yields the output feedback controller. The closed-loop system in the \( (x, e) \) coordinates has the block-triangular structure:

\[ \begin{pmatrix} \dot{x} \\ \dot{e} \end{pmatrix} = \begin{pmatrix} A - BK & BK \\ 0 & A - LC \end{pmatrix} \begin{pmatrix} x \\ e \end{pmatrix}. \]

(Separation Principle) The closed-loop eigenvalues are \( \sigma(A - BK) \cup \sigma(A - LC) \). Hence the state feedback and observer designs can be carried out independently. The combined controller stabilizes the system if and only if both \( A - BK \) and \( A - LC \) are Hurwitz.

Chapter 5: Poles and Zeros of MIMO Systems

Rational Matrices

A rational matrix \( \mathbf{H}(s) \in \mathbb{R}^{p \times m}(s) \) has entries that are ratios of polynomials. It is proper if \( \lim_{s \to \infty} \mathbf{H}(s) \) is finite. Poles and zeros of MIMO systems are more subtle than in SISO: individual entries may have different poles, and “zeros” have a system-theoretic interpretation.

Smith Form and Smith–McMillan Form

Smith Normal Form

For a polynomial matrix \( P(s) \in \mathbb{R}[s]^{p \times m} \) of rank \( r \), there exist unimodular matrices \( U(s) \) (invertible polynomial matrices with polynomial inverse) such that

\[ P(s) = U(s) \begin{pmatrix} \Sigma(s) & 0 \\ 0 & 0 \end{pmatrix} V(s), \]

where \( \Sigma(s) = \text{diag}(\sigma_1(s), \ldots, \sigma_r(s)) \) with monic polynomials \( \sigma_i \) satisfying \( \sigma_i \mid \sigma_{i+1} \). These \( \sigma_i \) are the invariant factors of \( P(s) \).

Smith–McMillan Form

Let \( \mathbf{H}(s) \in \mathbb{R}^{p \times m}(s) \) be a rational matrix of normal rank \( r \). Write \( \mathbf{H}(s) = N(s)/d(s) \) with \( d(s) \) the LCD of all entries. The Smith–McMillan form of \( \mathbf{H}(s) \) is \[ M(s) = \text{diag}\!\left(\frac{\epsilon_1(s)}{\psi_1(s)}, \ldots, \frac{\epsilon_r(s)}{\psi_r(s)}, 0, \ldots, 0\right), \]

where \( \epsilon_i, \psi_i \) are monic coprime polynomials, \( \epsilon_i \mid \epsilon_{i+1} \), and \( \psi_{i+1} \mid \psi_i \).

The Smith–McMillan form is unique and is related to the Smith form of \( N(s) \) via cancellation.

MIMO Poles and Zeros

The poles of \( \mathbf{H}(s) \) are the roots of \( \psi(s) = \prod_{i=1}^r \psi_i(s) \) (the pole polynomial). The transmission zeros are the roots of \( \epsilon(s) = \prod_{i=1}^r \epsilon_i(s) \) (the zero polynomial).

For a SISO system, these reduce to the ordinary poles and zeros of the scalar transfer function. For MIMO, a transmission zero at \( s = z \) means there exists a non-trivial direction in the input space that is blocked — the output becomes identically zero even though the input is nonzero at frequency \( z \). Algebraically, \( z \) is a transmission zero iff \( \mathbf{H}(z)v = 0 \) for some nonzero \( v \).

Poles vs. Eigenvalues of State-Space Realizations

For a realization \( (A,B,C,D) \), the eigenvalues of \( A \) are a superset of the poles of \( \mathbf{H}(s) \). The poles of \( \mathbf{H}(s) \) are the eigenvalues of \( A \) corresponding to the controllable-and-observable part of the system (i.e., the \( A_{co} \) block in the Kalman decomposition). In particular:

If the realization is minimal, then the poles of \( \mathbf{H}(s) \) equal the eigenvalues of \( A \).
If the realization is not minimal, some eigenvalues of \( A \) are cancelled in \( \mathbf{H}(s) \).

Minimal Realizations

A realization \( (A, B, C, D) \) of \( \mathbf{H}(s) \) is minimal if and only if it is both controllable and observable. All minimal realizations of the same transfer matrix are related by a similarity transformation.

The McMillan degree of \( \mathbf{H}(s) \) is \( \deg(\psi_1 \cdots \psi_r) \) and equals the order of any minimal realization.

Chapter 6: Introduction to Optimal Control

Motivation

Classical eigenvalue placement via pole assignment is purely qualitative — it places eigenvalues at desired locations but does not account for the cost of control effort or the magnitude of the state response. Optimal control seeks a systematic tradeoff: minimize a performance objective while stabilizing the system.

Linear Quadratic Regulator (LQR)

Consider the continuous-time LTI system \( \dot{x} = Ax + Bu \) with infinite-horizon cost

\[ J = \int_0^\infty \bigl(x(t)^\top Q x(t) + u(t)^\top R u(t)\bigr) dt, \]

where \( Q = Q^\top \succeq 0 \) and \( R = R^\top \succ 0 \) are weighting matrices. The matrix \( Q \) penalizes large states; \( R \) penalizes control effort.

Algebraic Riccati Equation

(LQR Theorem) Assume \( (A, B) \) is stabilizable and \( (Q^{1/2}, A) \) is detectable (equivalently, the pair \( (A, Q^{1/2}) \) has no unobservable unstable modes). Then there exists a unique positive semi-definite solution \( P = P^\top \succeq 0 \) to the algebraic Riccati equation (ARE) \[ A^\top P + PA - PBR^{-1}B^\top P + Q = 0. \]

The optimal control law is the linear state feedback

\[ u^*(t) = -Kx(t), \quad K = R^{-1}B^\top P, \]

and the minimum cost is \( J^* = x_0^\top P x_0 \). The closed-loop matrix \( A - BK = A - BR^{-1}B^\top P \) is Hurwitz.

Derivation via completing the square. For any stabilizing control \( u \), \[ \frac{d}{dt}(x^\top P x) = x^\top(A^\top P + PA)x + 2x^\top P Bu. \]

Using the ARE \( A^\top P + PA = PBR^{-1}B^\top P - Q \):

\[ \frac{d}{dt}(x^\top P x) = x^\top(PBR^{-1}B^\top P - Q)x + 2x^\top PBu. \]

Complete the square in \( u \):

\[ x^\top Q x + u^\top R u = x^\top Q x + u^\top R u - \frac{d}{dt}(x^\top P x) + x^\top(PBR^{-1}B^\top P - Q)x + 2x^\top P Bu \]\[ = \bigl(u + R^{-1}B^\top Px\bigr)^\top R \bigl(u + R^{-1}B^\top Px\bigr) - \frac{d}{dt}(x^\top Px). \]

Integrating from \( 0 \) to \( \infty \) (using \( x(\infty) = 0 \) for stabilizing controls):

\[ J = \int_0^\infty \bigl(u + Kx\bigr)^\top R \bigl(u + Kx\bigr) dt + x_0^\top P x_0. \]

Since \( R \succ 0 \), \( J \geq x_0^\top P x_0 \) with equality iff \( u = -Kx \).

Properties of LQR

For the SISO LQR problem (with \( m = p = 1 \)), the optimal closed-loop system has:

Gain margin: the loop gain can be reduced by a factor of 2 (6 dB) or increased to \( \infty \) without causing instability.
Phase margin: at least 60 degrees.
Disk margin: the return difference satisfies \( |1 + L(j\omega)| \geq 1 \) for all \( \omega \).

These “guaranteed margins” are a major practical advantage of LQR over arbitrary pole-placement.

Finite-Horizon LQR

For the finite-horizon problem with terminal cost, the optimal controller is time-varying: \( u^*(t) = -K(t)x(t) \) where \( K(t) = R^{-1}B^\top P(t) \) and \( P(t) \) satisfies the differential Riccati equation

\[ -\dot{P}(t) = A^\top P(t) + P(t)A - P(t)BR^{-1}B^\top P(t) + Q, \quad P(t_f) = Q_f, \]

integrated backward in time from the terminal condition \( P(t_f) = Q_f \succeq 0 \).

Output Feedback LQR

In practice the full state is unavailable; one observes \( y = Cx + v \) where \( v \) is measurement noise. The separation principle suggests using a state estimator (observer) to produce \( \hat{x}(t) \) and then applying \( u = -K\hat{x} \). The optimal estimator for stochastic noise is the Kalman filter, leading to the LQG design.

Minimum-Energy Estimation (Kalman Filter Motivation)

Consider the continuous-time system

\[ \dot{x} = Ax + Bu + w, \quad y = Cx + v, \]

where \( w \sim \mathcal{N}(0, W) \) (process noise, \( W \succeq 0 \)) and \( v \sim \mathcal{N}(0, V) \) (measurement noise, \( V \succ 0 \)) are independent white noise processes.

The steady-state Kalman filter is a linear observer of the form \[ \dot{\hat{x}} = A\hat{x} + Bu + L_f(y - C\hat{x}), \]

where the optimal gain \( L_f \in \mathbb{R}^{n \times p} \) minimizes the steady-state estimation error covariance \( \Sigma = \mathbb{E}[(x - \hat{x})(x-\hat{x})^\top] \).

(Continuous-Time Kalman Filter) Assume \( (A, W^{1/2}) \) is stabilizable and \( (C, A) \) is detectable. The optimal filter gain is \[ L_f = \Sigma C^\top V^{-1}, \]

where \( \Sigma = \Sigma^\top \succ 0 \) is the unique positive definite solution to the estimator algebraic Riccati equation

\[ A\Sigma + \Sigma A^\top - \Sigma C^\top V^{-1} C \Sigma + W = 0. \]

The error \( e = x - \hat{x} \) evolves as \( \dot{e} = (A - L_f C) e + \text{noise terms} \), and \( A - L_f C \) is Hurwitz.

The estimator Riccati equation is the dual of the LQR Riccati equation: substitute \( A \to A^\top \), \( B \to C^\top \), \( Q \to W \), \( R \to V \). This duality is not coincidental — the Kalman filter solves a minimum-energy estimation problem that is dual to the LQR problem.

Linear Quadratic Gaussian (LQG) Control

Combining the LQR state feedback gain \( K = R^{-1}B^\top P \) with the Kalman filter gain \( L_f = \Sigma C^\top V^{-1} \) yields the LQG controller:

\[ \dot{\hat{x}} = (A - BK - L_f C)\hat{x} + L_f y, \quad u = -K\hat{x}. \]

(LQG Separation Principle) Under the stated stabilizability and detectability assumptions, the LQG controller achieves the minimum expected cost \[ J_{LQG} = J_{LQR}^* + \text{tr}(P W) + \text{tr}(K^\top R K \Sigma), \]

and the closed-loop system is asymptotically stable with eigenvalues \( \sigma(A - BK) \cup \sigma(A - L_f C) \). The LQR gain \( K \) and the Kalman filter gain \( L_f \) can be designed independently (separation principle).

Unlike LQR, LQG does not guarantee robustness margins. It is possible to construct LQG designs with arbitrarily small gain and phase margins. This motivates robust design methods (LQG/LTR, \( H_\infty \) control) that are beyond the scope of this course but are natural extensions.

Introduction to Model Predictive Control (MPC)

Model Predictive Control (MPC) extends optimal control to handle constraints on states and inputs explicitly, at the cost of solving a finite-horizon optimization problem at each time step.

Basic Formulation

At time \( t \), given the current state \( x(t) \), solve the finite-horizon optimal control problem over a prediction horizon \( T_p \):

\[ \min_{u(\cdot)} \int_t^{t+T_p} \bigl(x(\tau)^\top Q x(\tau) + u(\tau)^\top R u(\tau)\bigr) d\tau + x(t+T_p)^\top P_f x(t+T_p) \]

subject to:

\( \dot{x}(\tau) = Ax(\tau) + Bu(\tau) \), \( x(t) = \text{current state} \),
\( x(\tau) \in \mathcal{X} \) (state constraints),
\( u(\tau) \in \mathcal{U} \) (input constraints).

Only the first piece of the optimal input is applied: \( u(t) = u^*(t | t) \). The problem is resolved at time \( t + \delta \) with updated state measurement (receding horizon strategy).

Key Properties

Constraint satisfaction: By design, input and state constraints are enforced.
Stability: With a suitable terminal cost \( P_f \) (often the LQR cost matrix) and terminal constraint set, MPC is guaranteed to be asymptotically stabilizing.
Computational complexity: For linear systems with convex constraints and quadratic cost, the optimization is a quadratic program (QP), solvable efficiently online.

Discrete-Time MPC

In practice, MPC is most commonly implemented in discrete time. Given \( x_k \), solve

\[ \min_{\{u_k, u_{k+1}, \ldots, u_{k+N-1}\}} \sum_{j=0}^{N-1} \bigl(x_{k+j}^\top Q x_{k+j} + u_{k+j}^\top R u_{k+j}\bigr) + x_{k+N}^\top P_f x_{k+N} \]

subject to \( x_{k+j+1} = A_d x_{k+j} + B_d u_{k+j} \), \( u_{k+j} \in \mathcal{U} \), \( x_{k+j} \in \mathcal{X} \). Apply only \( u_k \) and repeat.

Appendix: Summary of Key Results

State-Space Solution

\[ x(t) = e^{At}x_0 + \int_0^t e^{A(t-\tau)}Bu(\tau)\,d\tau, \quad \mathbf{H}(s) = C(sI-A)^{-1}B + D. \]

Lyapunov Equation

\( A \) is Hurwitz \( \iff \) \( A^\top P + PA + Q = 0 \) has a unique \( P \succ 0 \) for every \( Q \succ 0 \).

Controllability (PBH)

\( (A,B) \) controllable \( \iff \) \( \text{rank}[\lambda I - A, B] = n \) for all \( \lambda \in \mathbb{C} \).

Observability (PBH)

\( (C,A) \) observable \( \iff \) \( \text{rank}\begin{bmatrix}\lambda I - A \\ C\end{bmatrix} = n \) for all \( \lambda \in \mathbb{C} \).

LQR Riccati Equation

\( A^\top P + PA - PBR^{-1}B^\top P + Q = 0 \), \( K = R^{-1}B^\top P \), closed-loop: \( A - BK \) Hurwitz.

Kalman Filter Riccati Equation

\( A\Sigma + \Sigma A^\top - \Sigma C^\top V^{-1} C \Sigma + W = 0 \), \( L_f = \Sigma C^\top V^{-1} \), \( A - L_f C \) Hurwitz.

Smith–McMillan Form

Transfer matrix \( \mathbf{H}(s) \) has Smith–McMillan form \( \text{diag}(\epsilon_1/\psi_1, \ldots, \epsilon_r/\psi_r) \); poles are roots of \( \prod \psi_i \), transmission zeros are roots of \( \prod \epsilon_i \).

Separation Principle (LQG)

Eigenvalues of LQG closed-loop = \( \sigma(A-BK) \cup \sigma(A-L_f C) \); the two designs are independent.