Pole Placement

1. 49329 Control of Mechatronic Systems –Shoudong Huang Basic Properties and Pole Placement 1
49329 Control of Mechatronic Systems
Lecture 2
Basic Properties and Pole Placement
Dr. Shoudong Huang
Robotics Institute
Faculty of Engineering & IT
UTS

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Linear Continuous-Time
State Space Models
A continuous-time linear time-invariant state
space model takes the form
where x ∈ Rn is the state vector, u ∈ Rm is the
control signal, y ∈ Rp is the output, x0 ∈ Rn is the
state vector at time t = t0 and A, B, C, and D are
matrices of appropriate dimensions.
State space models use first order Ordinary Differential Equations (ODE)

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Some basic properties
State space model properties:
 Stability
 Controllability
 Observability

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Stability
is called asymptotically stable if for any initial state,
the state x(t) converges to zero as t increases indefinitely.
MATLAB demo
Assume u(t)=0 for all t, the system
or simply

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Stability
The system
is called asymptotically stable if for any initial state, the state
x(t) converges to zero as t increases indefinitely.
Condition of stability: all the eigenvalues of A have
negative real parts.
Eigenvalues of A are the solutions of the equation
Example:

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Matrices A and have the same eigenvalues.
Similarity Transformations
Similarity transformation does not change the stability of
the system:
where

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Similarity Transformations
Especially, matrix A can be diagonalized by a similarity
transformation T as
where λ1, λ2, …, λn are the eigenvalues of A.
For the new state-space model, because Λ is diagonal, we have
where the subscript i denotes the ith component of the state
vector. This component converges to 0 (when t increases) if λi
has negative real part.

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Example: Pendulum

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Example: Pendulum
MATLAB demo

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Controllability
An important question that lies at the heart of
control using state space models is:
Whether we can steer the state via the control input
to certain locations in the state space.
Technically, this property is called controllability.

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Controllability
The issue of controllability concerns whether a
given initial state x0 can be steered to the origin in
finite time using the input u(t).
Formally, we have the following:
Definition: A state x0 is said to be controllable
if there exists a finite interval [0, T] and an input
{u(t), t ∈ [0, T]} such that x(T) = 0. If all states are
controllable, then the system is said to be
completely controllable.

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A result from linear algebra
Cayley-Hamilton theorem: Every matrix satisfies
its own characteristic equation - i.e., if

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Test for Controllability
Theorem: The state space model
is completely controllable if and only if the matrix
has full row rank.
Proof: Uses Cayley-Hamilton Theorem – see notes.

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Example
Consider the state space model
The controllability matrix is given by
Clearly, its rank is 2; thus, the system is completely
controllable.

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Example
Consider the state space model
The controllability matrix is given by:
Its rank = 1 < 2; thus, the system is not completely
controllable.

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Observability
Observability is concerned with the issue of what
can be said about the state when one is given
measurements of the plant output.
A formal definition is as follows:
Definition: The system is said to be completely
observable if every state x(t0) = x0 can be
determined from the observation y(t) over a finite
time interval [t0 t1].

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Test for Observability
Theorem: Consider the state model
The system is completely observable if and only if the matrix,
has full column rank n.
Proof: Uses Cayley-Hamilton Theorem – see notes.

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Example
Consider the following state space model:
Then
Hence, its rank is 2, and the system is completely
observable.

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Example
Consider
Here
Hence, its rank=1 < 2, and the system is not
completely observable.

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Pole Placement (Pole Assignment)
We can design the closed-loop system to have good properties by
assigning its poles --- the eigenvalues of matrix A-BK.

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Pole Placement
MATLAB
K=place(A,B,P)
Demo

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Pole Placement
We can design the closed-loop system to have good properties by
assigning its poles --- the eigenvalues of matrix A-BK.
Necessary and sufficient condition for arbitrary pole placement
---- the system is completely controllable.
(see notes for proof)

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Summary
• Stability of the system can be studied from the eigenvalues of
the matrix A.
• State space models facilitate the study of certain system
properties that are paramount in the solution to the control
design problem. These properties relate to the following
questions:
 By proper choice of the input u, can we steer the system state to a desired state
(point value)? (controllability)
 If one knows the input, u(t), for t ≥ t0, can we infer the state at time t = t0 by
measuring the system output, y(t), for t ≥ t0? (observability)
• Pole placement is using feedback control law u=-Kx to place the
poles of the closed-loop system --- the eigenvalues of A-BK.