Linear–quadratic–Gaussian control

Key takeaways

Linear–quadratic–Gaussian control belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Linear–quadratic–Gaussian control to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Linear–quadratic–Gaussian control from memory before moving on to harder problems.

Reference excerpt

In control theory, the linear–quadratic–Gaussian (LQG) control problem is one of the most fundamental optimal control problems, and it can also be operated repeatedly for model predictive control. It concerns linear systems driven by additive white Gaussian noise. The problem is to determine an output feedback law that is optimal in the sense of minimizing the expected value of a quadratic cost criterion. Output measurements are assumed to be corrupted by Gaussian noise and the initial state, likewise, is assumed to be a Gaussian random vector. Under these assumptions an optimal control scheme within the class of linear control laws can be derived by a completion-of-squares argument. This control law which is known as the LQG controller, is unique and it is simply a combination of a Kalman filter (a linear–quadratic state estimator (LQE)) together with a linear–quadratic regulator (LQR). The separation principle states that the state estimator and the state feedback can be designed independently. LQG control applies to both linear time-invariant systems as well as linear time-varying systems, and constitutes a linear dynamic feedback control law that is easily computed and implemented: the LQG controller itself is a dynamic system like the system it controls. Both systems have the same state dimension. A deeper statement of the separation principle is that the LQG controller is still optimal in a wider class of possibly nonlinear controllers. That is, utilizing a nonlinear control scheme will not improve the expected value of the cost function. This version of the separation principle is a special case of the separation principle of stochastic control which states that even when the process and output noise sources are possibly non-Gaussian martingales, as long as the system dynamics are linear, the optimal control separates into an optimal state estimator (which may no longer be a Kalman filter) and an LQR regulator. In the classical LQG setting, implementation of the LQG controller may be problematic when the dimension of the system state is large. The reduced-order LQG problem (fixed-order LQG problem) overcomes this by fixing a priori the number of states of the LQG controller. This problem is more difficult to solve because it is no longer separable. Also, the solution is no longer unique. Despite these facts numerical algorithms are available to solve the associated optimal projection equations which constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller. LQG optimality does not automatically ensure good robustness properties. The robust stability of the closed loop system must be checked separately after the LQG controller has been designed. To promote robustness some of the system parameters may be assumed stochastic instead of deterministic. The associated more difficult control problem leads to a similar optimal controller of which only the controller parameters are different. It is possible to compute the expected value of the cost function for the optimal gains, as well as any other set of stable gains. The LQG controller is also used to control perturbed non-linear systems.

Mathematical description

Continuous time Consider the state-space representation of a continuous-time linear dynamic system

x ˙ ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) + v ( t ) , y ( t ) = C ( t ) x ( t ) + w ( t ) , {\displaystyle {\begin{aligned}{\dot {\mathbf {x} }}(t)&=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)+\mathbf {v} (t),\\\mathbf {y} (t)&=C(t)\mathbf {x} (t)+\mathbf {w} (t),\end{aligned}}}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Linear–quadratic–Gaussian control

Start with the simplest possible case. Write down what Linear–quadratic–Gaussian control claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Linear–quadratic–Gaussian control before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Linear–quadratic–Gaussian control ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Linear–quadratic–Gaussian control

In research: Linear–quadratic–Gaussian control appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Linear–quadratic–Gaussian control in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Linear–quadratic–Gaussian control is common in secondary-school and first-year university syllabi. It links to neighbouring topics Control loop theory, Optimal control, Stochastic control, so understanding it makes those chapters shorter.
In everyday life: Look for Linear–quadratic–Gaussian control outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Linear–quadratic–Gaussian control in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Linear–quadratic–Gaussian control means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Linear–quadratic–Gaussian control out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Linear–quadratic–Gaussian control in simple terms?

In control theory, the linear–quadratic–Gaussian (LQG) control problem is one of the most fundamental optimal control problems, and it can also be operated repeatedly for model predictive control. It concerns linear systems driven by additive white Gaussian noise.

Why does Linear–quadratic–Gaussian control matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Linear–quadratic–Gaussian control?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Linear–quadratic–Gaussian control.